DIFFERENTIAL 



AND 



Integral Calculus 

ON THE BASIS OF CONTINUOUS QUANTITY AND 
CONSECUTIVE DIFFERENCES. 



DESIGNED FOR 



ELEMENTARY INSTRUCTION, 



BY 

/ 

CHARLES DAVIES, LL.D., 

3XERITUS PROFESSOR OP HIGHER MATHEMATICS IN COLUMBIA COLLEGE. 



i c o 



NEW YORK * : ■ CINCINNATI • : • CHICAGO 

AMEKICAN BOOK COMPANY 



THE LIBRARY OF 

CONGRESS, 
Two Cowii RtewvED 

AUG. 23 1901 

~ Copyright ehtrv 
COPY B. 



Copyright, 1873, by Charles Davies 

Copyright, 1901, by J. Mansfield Davies, 

Eunice R. Davies Allen, and Alice Davies 



D. CAL. 

W. P. I 



. 1 



PKEFACE 



The Differential and Integral Calculus is too important a 
branch of Mathematics to be omitted in a course of col- 
legiate instruction. 

In the elementary branches, the abstract quantities. 
Number and Space, are presented to the mind as of defi- 
nite extent, and as made up of parts. 

The v ,lue, or measure, in any given case, is expressed by 
the number of times which the quantity contains one of 
its parts, regarded as a standard, or unit of measure. But 
we do not attain to a full and clear apprehension of their 
quantitative nature, until we subject them to the law of 
continuity, and trace their changes, under this law, as 
they pass from one state of value to another 

The Differential and Integral Calculus embraces all the 
processes necessary to such an analysis. It regards quan- 
tity as the result of change. It examines established 
laws of change, and determines their consequences. It 
supposes laws of change, and traces the results of the 
hypothesis. In short, it embraces within its grasp — in the 
Material, everything from the minutest atom to the largest 
body — in Space, all that can be measured, from the geo- 
metrical point to absolute infinity — in Time, the entire 
range of duration — and in Motion, every change from 
absolute rest to infinite velocity. 



17 PKEFACE. 

The substance of the present volume was published m 
the year eighteen hundred and sixty. For the want of 
a proper introduction, its marked characteristics seem not 
to have attracted public attention. That introduction is 
now supplied. 

The entire system is based on four principles: 

1st. Continuous quantity, which is defined; 

2d. Consecutive values of continuous quantity, which 
are defined with reference to the Calculus ; 

3d. The Differential of a quantity, arising from the two 
first definitions ; viz., the difference betiveen any two con- 
secutive values of a continuous quantity ; 

4th. That the differential of the independent variable 
is the common unit of measure for all differentials. 

The law of continuity, which may be applied, by hy- 
pothesis, to all quantity, not restricted by definition, and 
which is certainly applicable to time and space, and so 
far as we know, to all growth and development, is a 
necessary condition in all operations of the Calculus. 
Hence, the true theory of the Calculus must be based 
upon it. 

In regard to the present Treatise, it aims to make the 
law of continuity, in quantity, as accessible and familiar 
as the law of gravitation in matter. If this be accom- 
plished, the mysteries of the Calculus will disappear; 
and the subject will embrace, like the other branches 
of Mathematics, only questions of ratio and measure- 
ment. And why should this not be so, after we ap- 
prehend, distinctly, the unit of measure and the law 
of change? 

Fishkill-on- Hudson, January, 18T3. 



CONTENTS. 



INTRODUCTION. 

PAOK 

COHTINUOUS QUANTITY— CONSECUTIVE DIFFERENCES AND LIMITS, 9-44 



SECTION I. 

DEFINITIONS AND FIB8T PRINCIPLES 

iBTIOLB 

Definitions 1 

Uniform and Varying Changes 2-4 

Function and Variable 4-9 

Algebraic and Transcendental Functions ... 9 

Geometrical representation of Functions 10 

Language of Numbers inadequate 11 

Consecutive Values and Differentials 12 

Differential Coefficient . 13 

Form of difference between two states of a Function 14-15 

Differential Coefficient and Differential 15 

Equal Functions have equal Differentials 16 

Converse not True 17-19 

Signs of the Differential Coefficient 19 

Nature of a Differential Coefficient, and of a Differential 20 

Rate of Change 21-24 

Nature of Differential Calculus 2*4-26 



SECTION II. 

DIFFERENTIALS OF ALGEBRAIC FUNCTIONS. 

Differential of Sum or Difference of Functions 26 

Differential of a Product 27-29 

Differentials of Fractions 29 

Differentials of Powers and Formulas 30 

Differential of a Particular Binomial 30 

Rate of Change of a Function 31 

Partial Differentials 82-34 

14 



VI CONTENTS. 

section m. 

INTEGRATION AND APPLICATIONS. 

4.BTI0LJ 

Integration and A pplications 34 

Integration of Monomials 35-41 

Integration of Particular Binomials 41 

Integration by Series 42 

Equations of Tangents and Normals 43-50 

Asymptotes 50-52 

Differential of an Arc 52 

Rectification of Plane Curves 53 

Quadratures 54 

Quadrature of Plane Figures 55 

Nature of the Integral 56 

Area of a Rectangle , 67 

Area of a Triangle 58 

Area of a Parabola 59 

Area of a Circle 60 

Area of an Ellipse 61 

Quadrature of Surfaces of Revolution . . 62 

Surface of a Cylinder 63 

Surface of a Cone 64 

Surface of a Sphere 65 

Surface of a Paraboloid 66 

Surface of an Ellipsoid 67 

Cubature of Volumes of Revolution 68 

Examples in Cubature 68-7 1 



SECTION IV. 

SUCCESSIVE DIFFERENTIALS — STGNS OF DIFFERENTIAL CO- 
EFFICIENTS — FORMULAS OF DEVELOPMENT. 

Successive Differentials 71 

Signs of the First Differential Coefficient. . 72 

Signs of the Second Differential Coefficient 73 

Applications 74 

Maclaurin's Theorem , 75-76 

Taylors Theorem 77-81 



C O NTENTS. FU 



SECTION Y. 

MAXIMA AND MINIMA. 

ABTI0L3 

Maxima and Minima 81-86 

Points of Inflection 84 



SECTION VI. 

DIFFERENTIALS OF TRANSCENDENTAL FUNCTIONS. 

Differentials of Logarithmic Functions 86 

Relation between a and k 87-90 

Differential Forms which have Known Logarithmic Integrals 91 

Circular Functions 92-99 

Differential Forms which have Known Circular Integrals 99 

Applications 100 



SECTION VII. 

TRANSCENDENTAL CURVES — CURVATURE — RADIUS OF CURVA- 
TURE — INVOLUTES AND EVOLUTES. 

Classification of Curves 101 

Logarithmic Curve — General Properties 102-106 

Asymptote 106 

Sub-tangent 107 

The Cycloid 108 

Transcendental Equation of the Cycloid 109 

Differential Equation 110 

Sub-tangent — Tangent — Sub-normal — Normal Ill 

Position of Tangent 112 

Curve Concave .. . 113 

Area of the Cycloid 114 

Area of Surface generated by Cycloidal Arc 115 

Volume generated by Cycloid H6 

Spirals, or Polar Lines 117 

General Properties ' us 

Spiral of Archimedes 119 

Parabolic Spiral 120 

Hyperbolic Spiral 12] 



VJU CONTENTS. 

ABTfOU 

Logarithmic Spiral 122 

DirectioD of the Measuring Arc 1 23 

Sub-tangent in Polar Curves 124-127 

Angle of Tangent and Radius-vector 127 

Value of the Tangent 128 

Differential of the Arc 129 

Differential of the Area 130 

Areas of Spirals 131-135 

CU RVA TURE. 

Curvature of a Circle inversely as the Radius . . 135-136 

Orders of Contact 137 

Osculatory Curves 138 

Osculatory Circle 139 

Limit of the Orders of Contact 140 

Radius of Curvature 141 

Measure of Curvature 142-1 44 

Radius of Curvature for Lines of the Second Order 144-149 

Evolute Curves 149 

A Normal to the Involute is Tangent to the Evolute 150 

Evolute and Radius change by Same Quantity 151 

Evolute of the Cycloid 152 

Equation of the Evolute Curve 153 

Evolute of the Common Parabola 154 



INTEGRAL CALCULUS. 

Nature of Integration 155 

Forms of Differentials having known Algebraic Functions 156-159 

Forms of Differentials having known Logarithmic Functions 159 

Forms of Differentials having known Circular Functions 160 

Integration of Rational Fractions 161 

Integration by Parts 162 

Integration of Binomial Differentials 163 

When a Binomial can be Integrated 164 

Formula ^ 165 

Formula Jg 166 

Formula <g 167 

Formula J& 168 

Formula Jg 169 



INTRODUCTION 



TO 



DIFFERENTIAL CALCULUS. 



1. The entire science of mathematics is conversant about 
the properties, relations, and measurement of quantity. 
Quantity has already been defined. It embraces everything 
which can be increased, diminished, and measured. 

In the elementary branches of mathematics, quantity is 
regarded as made up of parts. If the parts are equal, each 
is called a unit, and the measure of a quantity is the num- 
ber of times which it contains its unit. Such quantities 
are called discontinuous ; because, in passing from one state 
of value to another, we go by the steps of the unit, and 
hence, pass over all values lying between adjacent units. 

Thus, if we increase a line from one foot to forty feet, by 
the continued addition of one foot, we touch the line, in oui 
computation, only at its two extremities, and at thirty-nine 
intermediate points, of which any two adjacent points are 
one foot apart. In the scale of ascending numbers, 1, 2, 3, 
4, 5, 6, etc., we pass over all quantities less than that which 
is denoted by the unit, one. Discontinuous quantities are 
generally expressed by numbers, or by letters, which stand 
for numbers. 



10 INTRODUCTION TO 

2. In the higher branches of mathematics, the laws which 
regulate and determine the changes of quantity, from one 
state of value to another, are quite different. Suppose, for 
example, that instead of considering a right line to be made 
up of forty feet, or of 480 inches, or of 960 half inches, or 
of 1,920 quarter inches, or of any number of equal parts of 
the inch, we regard it as a quantity having its origin at 0, 
and increasing according to such a law, as to pass through 
or assume, in succession, all values between and forty feet. 
This supposition gives us the same distance as before, but 
a very different law of formation. A quantity so formed or 
generated, is called a continuous quantity. Hence, 

A discontinuous quantity is one which is made up of 
parts, and in which the changes, in passing from one state 
of value to another, can be expressed in numbers, either 
exactly, or approximatively ; and 

A continuous quantity is one which in changing from 
one state of value to another, according to a fixed law, 
passes through or assumes, in succession, all the interme- 
diate values. 

Thus, the time which elapses between 12 and 1 o'clock, 
or between any two given periods, is continuous. All space 
is continuous, and every quantity may be regarded as con- 
tinuous, which can be subjected to the required law of 
change. 

Limits. 

3. The limit of a variable quantity is a quantity towards 
which it may be made to approach nearer than any given 
quantity, and which it reaches, under a particular supposi- 
tion. 



DIFFERENTIAL CALCULUS. 11 

Limits of Discontinuous Quantity. 

4. The limits of a discontinuous quantity are merely 
numerical boundaries, beyond which the quantity cannot 



For positive quantities, the minimum limit is 0, and the 
maximum limit, infinity. For negative quantities, they are 
0, and minus infinity; and generally, using the algebraic 
language, the limits of all quantities are, 

Minimum limit, — infinity ; maximum limit, + infinity. 

We can illustrate these limits, and also what we mean 
by the terms, and infinity, plus or minus, by refer- 
ence to the trigonometrical functions. Thus, when the arc 
is 0, the sine is 0. When the arc increases to 90°, the sine 
attains its maximum value, the radius, R. Passing into the 
second quadrant, the sine diminishes as the arc increases, 
and when the arc reaches 180°, the sine becomes 0. From 
that point, to 270°, the sine increases numerically, but de- 
creases algebraically, and at 270°, its minimum value is, 
— R. From 270° to 360°, the sine decreases numerically, 
but increases algebraically. Hence, the numerical limits of 
the sine, are and R; and its algebraic limits, — R and 
+ R. 

Let us now consider the tangent. For the arc 0, the 
tangent is 0. If the arc be increased from towards 90°, 
the length of the tangent will increase, and as the arc 
approaches 90°, the prolonged radius or secant becomes 
more nearly parallel with the tangent; and finally, at 90° 
it becomes absolutely parallel to it, and the length of the 



12 INTRODUCTION TO 

tangent becomes greater than any assignable line. Then 
we say, that the tangent of 90° is infinite; and we de- 
signate that quantity by oo . After 90°, the tangent becomes 
minus, and continues so to the end of the second quadrant, 
where it becomes — ; and at 270° it becomes equal to, + oo . 
The secant of 90° is also equal to + oc ; and of 270°, to — oo . 
These illustrations indicate the significations of the terms, 
zero and infinity. They denote the limits toward which varia- 
ble quantities may be made to approach nearer than any 
given quantity, and which limits are reached under particu- 
lar suppositions. 

5. The term, given, or assignable quantity, denotes any 
quantity of a limited and fixed value. 

The term, infinitely great, or infinity, denotes a quantity 
greater than any assignable quantity of the same kind. 

The term, infinitely small, or infinitesimal, denotes a 
quantity less than any assignable quantity of the same 
kind. 

Continuous Quantities. 

6. A continuous quantity has already been defined 
(Art. 2.) By its definition it has two attributes : 

1st. That it shall change its value according to a fixed 
law; and 

2d. That in changing its value, between any two limits, 
it shall pass through all the intermediate values. 

7. Consecutive Values. — Two values of a continuous 
quantity are consecutive when, if the greater be diminished 



DIFFERENTIAL CALCULUS. 



13 



or the less increased, according to the law of change, the 
two values will become equal. 

Let A be the origin of a system of rectangular co-ordi- 
nate axes, and C a given point on the axis of X. 

If we suppose a point to move from A, in the plane of the 
axes, and with the further condition, that it shall continue 
at the same distance from the point C, it will generate the 
circumference of a circle, APBDEA, beginning and ter- 
minating at the point A. The moving point is called the 
generatrix. 

The circumference of this circle may also be generated in 
another way, thus : 

Denote the straight line AD by 272, and suppose a point 
to move uniformly from A to D. Denote the distance from 
A to any point 
of the line A D, 
by x : then, the 
other segment 
will be denoted 
by 2R - x. 
Now, at every 
point of AD, 
suppose a per- 
pendicular to 
be drawn to 
AD. Denote 
each perpendic- 
ular by y, and suppose y always to have such a value as to 
satisfy the equation 

if - 2Rx - x\ 




14 IKTRODUCTION TO 

Under these hypotheses, it is plain that the extremities of 
the ordinate y will be found in the circumference of the 
circle, which will be a continuous quantity. The ordinate 
y will be contained, in the first quadrant, between the 
numerical limits of y ~ and y = + R; in the second, be- 
tween the numerical limits of y = + R, and y = ; in the 
third, between y = — and y = — R; and in the fourth, 
between y = —R and y = — 0. 

The circumference ABDEA, may be regarded under two 
points of view : 

First. As a discontinuous quantity, expressed in numbers : 
yiz., by AD X 3.1416 ; or it may be expressed in degrees, 
minutes, or seconds, viz., 360°, or 21600', or 1296000". In 
the first case, the step, or change, in passing from one value 
to the next, will be the unit of the diameter AD. In the 
second, it will be one degree, one minute, or one second. In 
neither case, will the parts of the circumference less than the 
unit be reached by the computation. Or, 

Secondly: We may regard the circumference as a con- 
tinuous quantity, beginning and terminating at A. Under 
this supposition, the generatrix will occupy, in succession, 
every point of the circumference, and will, in every position, 
satisfy the equation 

%f = 2Rx - x\ 
Hence, if we measure a quantity by a finite unit, that 
quantity is discontinuous: but if we measure it by an infin- 
itesimal unit, the quantity becomes continuous. 

Tangent Line and Limit. 

§. Take any point of the circumference of this circle, as 
P, whose co-ordinates are x' and y', and a second point /f 



DIFFERENTIAL CALCULUS. 



15 



whose co-ordinates are x" and y", and through these points 
draw the secant line, RPG. 

Then, HJ= y" - y', and PJ= x" -x' ; and 



PJ x"-x' 



tang, of the angle HP J, or HGC. 



Let us now suppose a tangent line TP to be drawn to the 
circle, touching it at P. If we suppose the point H to ap- 
proach the point P, it is plain that the value of y" will ap- 
proach to the value of y', and the value of x" to that of x' : 
and when the point H becomes consecutive with the point 
P, y" and y' will become consecutive, and so also will x" and 
x\ When the point H becomes consecutive with the point 
P, the secant line, HG, becomes the tangent line PT. For, 
since the arc is a continuous quantity, no point of it can lie 
between two of its consecutive values ; and hence, at P, no 
point of the curve can lie above the line TP ; therefore, by 
the definitions of Geometry, TP is a tangent line to the circle 
at the point P. 

But the defini- 
tion of a tangent 
line to a circle, 
in elementary 
Geometry, viz., 
that it touches 
the circumfer- 
ence in one point 
is incomplete. It 
is provisional on- 
ly. For, as we 
now see, the tan- 




16 INTRODUCTION TO 

gent line touches the circle in tivo consecutive poitits, which, 
in discontinuous quantity, are regarded as one, because the 
distance between them, expressed numerically, is zero. 

If we prolong JHtill it meets the tangent line at 0, we see 
that, 

■ f = tangent of OPJ = tangent of OTC; and that, 



x" - x 

JH 
x" - x' 



tangent of HP J = tangent of HGC 



When the point H approaches the point P nearer than 
any given distance, the angle HGC will approach the angle 
PTC nearer than any given angle, and when i?" becomes con- 
secutive with P, the angle HG C will become equal to the 
angle PTC, which is therefore its limit. Under this hy- 
pothesis, the point H falls on the tangent line, and JH be- 
comes equal to JO. Under the same hypothesis, y" and y' 
become consecutive, and also x" and x' ; hence, y n —y' 
becomes less than any given quantity ; and so, also, does 
x" — x\ This difference between consecutive values is ex- 
pressed by simply writing the letter d before the variable. 
Thus, the difference of the consecutive values of y, is 
denoted by dy ; and is read differential of y ; and the dif- 
ference between the consecutive values of x 9 is denoted by dx, 
and is read differential of x. Hence, we have 

^ = tangent PTC; viz., 

the tangent of the angle which the tangent at the point P 
makes the axis of X. 



DIFFEKENTIAL CALCULUS. 



17 





Y 




( 


9 






Pr^ 


H 

J >v 




^/ 




V 




G 


T A 


\oo 


' a?" 


G -^ \ 



By the definition of a limit, cly becomes the limit of y" —y\ 
and dx the limit 

of x" — x\ under 
the supposition, 
that y" and y\ 
and z" and x f be- 
come, respective- 
ly, consecutive. 
The term limit, 
therefore, used to 
designate the ul- 
timate difference 
between two val- 
ues of a variable, 
denotes the actual difference between its two consecutive 
values; this difference is infinitely small, and consecutive 
with zero. For, if after y" has become consecutive with y\ 
it be again diminished, according to the law of change ex- 
pressed by the equation 

if =± 2Rx - x* 

it will, from the definition of consecutive values, become 
equal to y\ and then x" will become equal x', and we shall 
have 

x" - x' = 0. 



E 



y 



'r */ — 



y' = 



and 



Under this hypothesis the line FT has, at P, but a single 
point, common with the circumference of the circle ; it then 
ceases to be a tangent, and becomes any secant line passing 
through this point and intersecting the circumference in a 
second point. 

9. What we have here shown in regard to the circum- 

1* 



18 INTRODUCTION TO 

ference of the circle, and its tangent, is equally true of any 
other curve and its tangent, as may he shown by a very 
slight modification of the process. 

The fact, that a straight line tangent to a curve, has two 
consecutive points common with it, appears in all the ele- 
mentary problems of tangents. These conditions, are, an 
equality between the co-ordinates of the point of contact and 
the first differential co-efficients, at the same point, of the 
straight line and curve. These conditions fix the consecutive 
points common to the straight line and curve. 

Analysis, therefore, by its searching and microscopic 
powers — by looking into the changes which take place in 
quantity, as it passes from one state of value to another, 
develops properties and laws which lie beyond the reach of 
the numerical language. Thus, the distance between two 
consecutive points, on the circumference of a circle, cannot 
be expressed by numbers ; for, however small the number 
might be, chosen to express such a distance, it could be 
diminished, and hence, there would be intermediate points. 

The introduction, therefore, of continuous quantity, into 
the science of mathematics, brought with it new ideas and 
the necessity of a new language. Quantity, made up of parts, 
and expressed by numbers, is a very different thing from the 
continuous quantity treated of in the Differential and Integral 
Calculus. Here, the law of continuity, in the change from 
one state of value to another, is the governing principle, 
and carries with it many consequences. 

Time and space are the continuous quantities with which 
we are most conversant. If we take a moment in time, and 
look back to the past, or forward to the future, there is no 



DIFFEKENTIAL CALCULUS. 19 

interruption. The law of continuity is unbroken, and the 
infinite opens to our contemplation. If we take a point in . 
space, and through it conceive a straight line to be drawn, 
the law of continuity is also there, and the imagination runs 
along it, to the infinite, in either direction. The attraction 
of gravitation is a continuous force; and all the motions to 
which it gives rise, follow the law of continuity. All 
growth and development, in the vegetable and animal king- 
doms, so far as we know, conform to this law. This, there- 
fore, is the great and important law of quantity, and the 
Higher Calculus is conversant mainly about its development 
and consequences. 

Consequences of the Law of Continuity. 

1. The most striking consequence of the law of conti- 
nuity, is the fact, that whatever be the quantity subjected 
to this law, or whatever be the law of change, the differ- 
ence between any two of the consecutive values is an infini- 
tesimal, and hence cannot be expressed by numbers. 

2. Since a continuous quantity may be of any value, and 
be subjected to any law of change, the infinitesimal which 
expresses the difference between any two of its consecutive 
values, is a variable quantity; and hence, may have any 
value between zero and its maximum limit. 

3. The law of continuity in quantity, therefore, intro 
duces into the science of mathematics a class of variables 
called infinitesimals, or differentials. Every variable quan- 
tity has, at every state of its value, an infinitesimal corre- 
sponding to it. This infinitesimal is the connecting link, in 
the law of continuity, and will vary with the value of the 
quantity and the law of change. 



20 



INTKODUCTION TO 



4. In the Infinitesimal Calculus, the properties, relations, 
and measurement of quantities are developed by considering 
the laws of change to which they are subjected. The 
elements of the language employed, are symbols of those 
infinitesimals. 



NEWTON'S METHOD OF TREATING CONTINUOUS QUANTITY.* 

Lemma I. 
10. Quantities, and the ratios of quantities, which in any finite 

time converge continually to equality, and before the end of 

that time approach nearer, the one to the other, than by any 

given difference, become ultimately equal. 

If you deny it, suppose them to be ultimately unequal, 
and let D be their ultimate difference. Therefore, they 
cannot approach nearer to equality than by that given dif- 
ference D ; which is against the supposition. 

Lemma II. 
If in any figure, AacE, terminated by right lines A a, 
AE, and the curve acE, there be inscribed any number of 
parallelograms Ab 9 Be, Od, etc., comprehended under the 
equal bases, AB, BC, CD, etc., and the sides Bb, Cc, Dd, 
etc., jwrallel to one side Aa of the 
figure; and the parallelograms 
aEbl, bLcm, cMdn, dDEo are com- 
pleted; then, if the breadth of 
these parallelograms be supposed to 
be diminished, and their number 
to be augmented in finitum; I say, 
that the ultimate ratios ivhich the 
inscribed figure AKbLcMdD. the 




* Principia, Book I., Section I. 



DIEFEKEKTIAL CALCULUS. 



21 



circumscribed figure AalbmcndoE and the curvilinear figure 
AdbcdE will have to one another, are ratios of equality. ■ 

For, the difference of the inscribed and circumscribed 
figures is the sum of the parallelograms, Kl, Lm, Mn, Do, 
that is. (from the equality of their bases), the rectangle 
under one of their bases Kb and the sum of their altitudes 
Aa; that is, the rectangle ABla. But this rectangle, be- 
cause its breadth AB is supposed diminished in finitum, 
becomes less than any given space. And therefore, (by 
Lemma I.) the figures inscribed and circumscribed, become 
ultimately equal one to the other; and much more will the 
intermediate curvilinear figure be ultimately equal to either. 



Lemma III. 

11. The same ultimate ratios are also ratios of equality, 
when the breadths AB, BC, DO, etc., of the parallelograms 
are unequal and are all diminished in finitum. 

For, suppose AF to be the greatest breadth, and complete 
the parallelogram FAafi This par- 
allelogram will be greater than the 
difference of the inscribed and cir- 
cumscribed figures ; but because 
its breadth AF is diminished in 
finitum, it will become less than 
any given rectangle. 

Coe. 1. Hence, the ultimate sum 
of these evanescent parallelograms 
will, in all parts, coincide with the curvilinear figure. 




Cor. 2. Much more will the rectilinear figure compre 



22 



IHTRODUCTI ONTO 



hended under the chords of the evanescent arcs, ab, be, cd } 
etc. ultimately coincide with the curvilinear figure. 

Cor. 3. And also, the circumscribed rectilinear figure 
comprehended under the tangents of the same arcs. 

Cor. 4. And therefore, these ultimate figures (as to their 
perimeters, acE) are not rectilinear, but curvilinear limits 
of rectilinear figures. 

Lemma IV. 

12. If in huo figures, AacE, PprT, you inscribe (as before) 
tivo ranks of parallelograms, an equal number in each ranJc, 
and, where their breadths are diminished, in finitum, the 
ultimate ratios of the parallelograms in one figure to those 
in the other, each to each respectively, are the same ; I say, 
that those t too figures, AacE, PprT, are to one another in 
that same ratio. 



a 



X* 



E 




For, as the parallelograms in the one figure are severally 
to the parallelograms in the other, so (by composition) is 
the sum of all in the one to the sum of all in the other ; 
and so is the one figure to the other ; because (by Lemma 
III.), the former figure to the former sum, and the lattei 
figure to the latter sum, are both in the ratio of equality. 



DIFFERENTIAL CALCULUS. 23 

Cob. Hence, if two quantities of any kind are anyhow 
divided into an equal number of parts, and those parts, 
when their number is augmented, and their magnitude 
diminished in finitti?n,ha,ve a given ratio one to the other, 
the first to the first, the second to the second, and so on in 
order, the whole quantities will be one to the other in that 
same given ratio. For if in the figures of this lemma, the 
parallelograms are taken one to the other in the ratio of the 
parts, the sum of the parts will always be as the sum of the 
parallelograms ; and therefore, supposing the number of the 
parallelograms and parts to be augmented, and their magni- 
tudes diminished infinitum, those sums will be in the ulti- 
mate ratio of the parallelogram in the one figure to the cor- 
responding parallelogram in the other; that is (by the 
supposition), in the ultimate ratio of any one part of the 
one quantity to the corresponding part of the other. 

Lemma V. 

13. In similar figures all sorts of homologous sides, whether 
curvilinear or rectilinear, are proportional; and their 
areas are in the duplicate ratio of their homologous sides. 

Lemma VI. 

14. If the arc A CB, given in position, is subtended by the 
chord AB, and in any point A 
in the middle of the continued 
curvature, is touched ~by a right 
line AB, produced loth ways ; 
then, if the points A and B 
approach one another and 
meet, \become consecutive 1 ^ I 
my, the angle BAD contained between the chord and the tan- 
gent willbe diminished in finitum, and ultimately zvill vanish. 




u 



INTRODUCTION TO 



For, if it does not vanish, the arc A CB, will contain with 
the tangent AD, an angle equal to a rectilinear angle ; and 
therefore, the curvature at the point A will not be con- 
tinued, which is against the supposition. 

Lemma VII. 

15. The same thing being supposed, I say that the ultimate 

ratio of the arc, chord, and tangent, any one to any other, 

is the ratio of equality. 

For, while the point B approaches towards the point A, 
consider always AB and AD as produced to the remote 
points b and d, and parallel to the secant BD draw bd: 
and let the arc Acb be always similar to the arc ACB. 
Then, supposing the points A 
and B to coincide, [become 
consecutive], the angle dAb 
will vanish, by the preceding 
lemma; and therefore, the 
right lines A b, Ad (which 
were always finite), and the 
intermediate arc Acb, will co- 
incide, and become equal 

among themselves. Wherefore, the right lines AB, AD, 
and the intermediate arc ACB (which are always pro- 
portional to the former), will vanish, and ultimately acquire 
the ratio of equality. 

Cor. 1. Whence, if through B we draw BF parallel to 
the tangent, always cut- A_ D 

ting any right line AF 
passing through i and 
F. this line BF will be, 
ultimately, in the ratio of equality with the evanescent 






DIFFERENTIAL CALCULUS. 25 

arc ACB; because, completing the parallelogram AFBD, it 
is always in the ratio of equality with AD. 

Cor. 2. And if through B and A more right lines be 
drawn BE, BD, AF, A G, 

cutting the tangent AD a. e\ \ l) 

and its parallel BF, the 
ultimate ratio of the ab- 
scissas AD, AE, BF, BG, 
and of the arc AB, any 
one to any other, will be the ratio of equality. 

Cor. 3. And therefore, in any reasoning about ultimate 
ratios, we may freely use any one of those lines for any 
other. 



16. Scholium. — Those things which have been demonstrated 
of curve lines, and the superficies which they comprehend, may 
be easily applied to the curve superficies, and contents of solids. 
These lemmas are premised to avoid the tediousness of de- 
ducing perplexed demonstrations ad absurchim, according to 
the method of the ancient geometers. For demonstrations 
are more contracted by the method of indivisibles : but be- 
cause the indivisibles seem somewhat harsh, and therefore, 
that method is reckoned less geometrical, I chose rather to 
reduce the demonstrations of the following propositions to 
the first and last sums and ratios, of nascent and evanescent 
quantities ; that is, to the limits of those sums and ratios ; 
and so, to premise, as short as I could, the demonstration of 
those limits. For, hereby the same thing is performed as 
by the method of indivisibles; and now those principles 
being demonstrated, we may use them with more safety. 



26 INTRODUCTION TO 

Therefore, if hereafter I should happen to consider quanti- 
ties as made up of particles, or should use little curve lines 
for right ones, I would not be understood to mean indivisibles, 
but evanescent divisible quantities; not the sums and ra- 
tios of determinate parts, but always the limits of sums and 
ratios ; and that the force of such demonstrations always 
depends on the method laid down in the foregoing lemmas. 
Perhaps it may be objected, that there is no ultimate 
proportion of evanescent quantities; because the proportion, 
before the quantities have vanished, is not the ultimate, and 
when they are vanished, is none. But by the same argument 
it may be alleged, that a body arriving at a certain place, 
and there stopping, has no ultimate velocity ; because, the 
velocity, before the body comes to the place, is not its ulti- 
mate velocity ; when it has arrived, is none. But the answer 
is easy; for, by the ultimate velocity is meant, that with 
which the body is moved, neither before it arrives at its last 
place and the motion ceases, nor after ; but, at the very in- 
stant it arrives; that is, that velocity with which the body 
arrives at its last place, and with which the motion ceases. 
And in like manner, by the ultimate ratio of evanescent 
quantities is to be understood the ratio of the quantities 
not before they vanish, not afterwards, but with which they 
vanish. In like manner, the first* ratio of nascent quanti- 
ties is that with which they begin to be. And the first or 
last sum, is that with which they begin to be (or to be 
augmented or diminished). There is a limit which the velo- 
city at the end of the motion may attain, but not exceed. 
This is the ultimate velocity. And there is the like limit 
in all quantities and proportions that begin and cease to be. 
And since such limits are certain and definite, to determine 
the same is a problem strictly geometrical. But whatever is 



DIFFERENTIAL CALCULUS. 2? 

geometrical we may be allowed to use in determining and 

demonstrating any other thing that is likewise geometrical. 

* * * * * * 



Fruits of Newton's Theory. 

17. The main difficulties in the higher mathematics, have 
arisen from inadequate or erroneous notions of ultimate or 
evanescent quantities, and of the ratios of such quantities. 
After two hundred years of discussion, of experiment and of 
trial, opinions yet differ widely in regard to them, and es- 
pecially in regard to the forms of language by which they 
are expressed. 

One cannot approach this subject, which has so long 
engaged the earnest attention of the greatest minds known 
to science, without a feeling of awe and distrust. But 
tapers sometimes light corners which the rays of the sun 
do not reach ; and as we must adopt a theory in a system of 
scientific instruction, it is perhaps due to others, that we 
should assign our reasons therefor. 

18. An ultimate, or evanescent quantity, which is the 
basis of the Newtonian theory, is not the quantity " be- 
fore it vanishes, nor afterwards ; but, ivith which it 
vanishes" 

I have sought, in what precedes and follows, to define 
this quantity — to separate it from all other quantities — 
to present it to the mind in a crystallized form, and in 
a language free from all ambiguity; and then to explain how 
it becomes the key'of a sublime science. 



28 INTRODUCTION TO 

As a first step in this process, I have defined continuous 
quantity (Art. 2.), and this is the only class of quantity to 
which the Differential Calculus is applicable. The next step 
was to define consecutive values, and then, the difference be- 
tween any two of them (Art. 7). These differences are the 
ultimate or evanescent quantities of Newton. They are 
not quantities of determinate magnitudes, but such as 
come from variables that have been diminished indefi- 
nitely. They form a class of quantities by themselves, 
which have their own language and their own laws of 
change ; and they are called, Infinitesimals, or Differentials. 

Since the difference between any two values of a variable 
quantity, which are near together, but not consecutive, will 
depend on the relative values of the quantities and the law 
of change, it is plain, that when we pass to the limit of this 
difference, such limit will also depend for its value on 
the variable quantity and the law of change : and hence, 
the infinitesimals are unequal among themselves, and any 
two of them may have, the one to the other, any ratio 
whatever. 

These infinitesimals will always be quantities of the same 
hind as those from which they were derived ; for the kind of 
quantity which expresses a difference, is the same, whether 
the difference be great or small. 

Limits. 

19. Marked differences of opinion exist among men of 
science in regard to the true notion of a limit ; and hence, 
definitions have been given of it, differing widely from each 
other. We have adopted the views of Newton, so clearly set 
forth in the lemmas and scholium which we have quoted 



DIFFERENTIAL CALCULUS. 



29 



from the Principia. He uses, as stated in the latter part 
of the scholium, the term limit, to designate the ultimate 
or evanescent value of a variable quantity; and this value is 
reached under a particular hypothesis. Hence, our defini- 
tion (Art. 3). 

Let us now refer again to the case of tangency. 

Let APB be any curve whatever, and TPF a tan- 
gent touching it at the point P. Draw any chord of 
the curve, as PB, and though P and B draw the or- 
dinates PD and BH. Also draw PC parallel to TH. 



Then, ^= tang. FPO : 
PC 



tang, the angle PTH, which 



the tangent line TPF makes with the axis TH. 
BC 



But, 
BPC. 



PC 



tangent of the angle 




A D 



If now we suppose BH to move 
towards PD, the angle BPC will 
approach the angle FPC, which 
is its limit. When BH becomes 
consecutive with PD, BC will 
reach its ultimate value : and since 
by Lemma VII., the uitimate 

ratio of the arc, chord, and tangent, any one to any other 
is the ratio of equality, it follows that they must then all be 
equal, each to each. Under this hypothesis the point B 
must fall on the tangent line TPF; that is, the chord and 
tangent, in their ultimate state, have two points in common ; 
hence they coincide ; and as the two points of the arc are 
consecutive, it must also coincide with the chord and tangent 



30 INTRODUCTION TO 

This, at first sight, seems impossible. But if it be granted 
that two points of a curve can be consecutive and that a 
straight line can be drawn through any two points, we have 
the solution. If we deny that two points of the curve can be 
consecutive, we deny the law of continuity. 

The method of Leibnitz adopted the simple hypothesis 
that when the point B approached the point P, infinitely 
near, the lines (TFand CB become infinitely small, and that 
then, either may be taken for the other ; under which 
hypothesis the ratio of PC to OB, becomes the ratio of PC 
to CF. 

What the Lemmas of Newton prove. 

20. The first lemma, which is "the corner-stone and 
support of the entire system," predicates ulti?nate equality 
between any two quantities which continually approach 
each other in value, and under such a law of change, that, 
in any finite time they shall approach nearer to each other 
than by any given difference. The common quantity towards 
w r hich the quantities separately converge, is the limit of each 
and both of them, and this limit is always reached under a 
particular supposition. 

Lemmas II., III., and IV. indicate the steps by which we 
pass from discontinuous to continuous quantity. They 
introduce us, fully, to the law of continuity. They dem- 
onstrate the great truth, that the curvilinear space is the 
common limit of the inscribed and circumscribed parallelo- 
grams, and that this limit is reached under the hypothesis 
that the breadth of each parallelogram is infinitely small, 
and the number of them, infinitely great. Thus, we reach 



DIFFERENTIAL CALCULUS. 31 

the law of continuity; and each parallelogram becomes a 
connecting link, in passing from one consecutive value to 
another, when we regard the curvilinear area as a variable. 
That there might be no misapprehension in the matter, 
corollary 1, of Lemma III., affirms, that, " the ultimate sum 
of these evanescent parallelograms, will, in all parts, coin- 
cide with the curvilinear figure." Corollary 4, also, affirms 
that, " therefore, these ultimate figures (as to their perimeters, 
acE), are not rectilinear, but curvilinear limits of rectilinear 
figures : " that is, the curvilinear area AEa is the common 
limit of the inscribed and circumscribed parallelograms, 
and the curve Eclcia, the common limit of their perimeters. 
This can only take place when the ordinates, like Dd> Cc, 
Bk, become consecutive ; and then, the points o, n, m, and 
I fall on the curve. 

The law of continuity carries with it, necessarily, the 
ideas of the infinitely small and the infinitely great. These 
are correlative ideas, and in regard to quantity, one is the 
reciprocal of the other. The inch of space, as well as the 
curved line, or the curvilinear surface of geometry, has 
within it the seminal principles of this law. 

If we regard it as a continuous quantity, having in- 
creased from one extremity to the other, without missing any 
point of space, we have, the law of change, the infinitely 
small (the difference between two consecutive values, or the 
link in the law of continuity), and the infinitely great, in 
the number of those values which make up the entire 
line. 

It has been urged against the demonstrations of the 
lemmas, that a mere inspection of the figures proves the 
demonstrations to be wrong. For, say the objectors, there 



32 INTRODUCTION TO 

will be, always, obviously, "a portion of the exterior par 
allelograms lying without the curvilinear space." This is 
certainly true for any finite number of parallelograms. 

But the demonstrations are made under the express 
hypothesis, that, "the breadth of these parallelograms be 
supposed to be diminished, and their number to be aug- 
mented, infinitum" Under this supposition, as we have 
seen, the points, o, n, m, and I, fall in the curve, and then 
the areas named are certainly equal. 

Newton's Method in harmony with that of Leibnitz. 

21. The method of treating the Infinitesimal Calculus, 
by Leibnitz, subsequently amplified and developed by the 
Marquis L'Hopital, is based on two fudamental proposi- 
tions, or demands, which were assumed as axioms. 

I. That if an infinitesimal be added to, or subtracted 
from, a finite quantity, the sum or difference will be the 
same as the quantity itself. This demand assumes that the 
infinitesimal is so small that it cannot be expressed by 
numbers. 

II. That a curved line may be considered as made up of 
an infinite number of straight lines, each one of which is 
infinitely small. 

It is proved in Lemma II. that the sum of the ultimate 
rectangles Ab, Be, Cd, Do, etc., will be equal to the curvi- 
linear area AaE. This can only be the case when each is 
" less than any given space," and their number infinite. 
What is meant by the phrase, " becomes less than any 
given space ? " Certainly, a space too small to be expressed 
by numbers ; for, if we have such a space, so expressed, we 
can diminish it by diminishing the number, which would 






>-m 



DIFFERENTIAL CALCULUS. 33 

be contrary to the hypothesis. This ultimate value, then 
of either of the rectangles, is numerically zero: and hence, 
its addition to, or subtraction from, any finite quantity, 
would not change the value. The ultimates of Newton, 
therefore, conform to the first demand of Leibnitz, as 
indeed they should do ; for, they are not numerical quanti- 
ties, but connecting links in the law of continuity. 

It is proved in lemma VII., that the ultimate ratio of the 
arc, chord, and tangent, any one to any other, is the ratio of 
equality : hence, their ultimate values are equal. When this 
takes place, the two extremities of the chord become con 
secutive, and the remote extremity of the tangent falls on 
the curve, and coincides with the remote extremity of the 
chord: that is, F falls on the curve, and PB and PF, 
coincide with each other, and with the curve. The length 
of this arc, chord, or tangent, in their ultimate state, is 



Vdx* + dy% 

a value familiar to the most superficial student of the 
Calculus. 

Behold, then, one side of the inscribed polygon, when 
such side is infinitely small, and the number of them 
infinitely great. 

That such quantities as we have considered, have a con- 
ceivable existence as subjects of thought, and do or may 
have, proximatively, an actual existence, is clearly stated in 
the latter part of the scholium quoted from the Principia* 
It is there affirmed: "This is the ultimate velocity. And 
there is a like limit in all quantities and proportions which 
begin and cease to be. And since such limits are certain 
and definite, to determine the same is a problem strictly geo- 

2 



34 INTRODUCTION TO 

metrical. But whatever is geometrical we may be allowed 
to use in determining and demonstrating any other thing 
that is likewise geometrical." * Hence, the theory of New- 
ton conforms to the second demand in the theory of Leib- 
nitz. 

Different Definitions of a Limit. 

22. The common impression that mathematics is an 
exact science, founded on axioms too obvious to be disputed, 
and carried forward by a logic too luminous to admit of 
error, is certainly erroneous in regard to the Infinitesimal 
Calculus. From its very birth, about two hundred years, 
ago, to the present time, there have been very great differ- 
ences of opinion among the best informed and acutest minds 
of each generation, both in regard to its fundamental prin- 
ciples and to the forms of logic to be employed in their de- 
velopment. The conflicting opinions appear, at last, to have 
arranged themselves into two classes; and these differ, 
mainly, on this question: What is the correct apprehension 
and right definition of the word limit ? All seem to agree 
that the methods of treating the Calculus must be governed 
by a right interpretation of this word. The two definitions 
which involve this conflict of opinion, are these: 

1. The limit of a variable quantity is a quantity towards 
which it may be made to approach nearer than any given 
quantity and which it reaches under a particular supposition. 

And the following definition, from a work on the Infini- 
tesimal Calculus by M. Duhamel, a French author of recent 
date: 

* Note,— The italics are added ; they are not in the text. 



DIFFERENTIAL CALCULUS. 3,1 

2. The limit of a variable is the constant quantity which the 
variable indefinitely approaches, Met never reaches. 

This definition finds its necessary complement in the fol- 
lowing definition by the same author : 

"We call," says he, "an infinitely small quantity, or sim- 
ply, an infinitesimal, every variable magnitude of which tJie 
limit is zero. 

The difference between the two definitions is simply this: 
by the first, the variable, ultimately, reaches its limit; by 
the second, it approaches the limit, but never reaches it. 
This apparently slight difference in the definitions, is the 
dividing line between classes of profound thinkers; and 
whoever writes a Calculus or attempts to teach the subject, 
must adopt one or the other of these theories. The first is 
in harmony with the theories of Leibnitz and Newton, which 
do not differ from each other in any important particular. 
It seems also to be in harmony with the great law r s of quan- 
tity. In discontinous quantity, especially, we certainly in- 
clude the limits in our thoughts, and in the forms of our 
language. When we speak of the quadrant of a circle, we 
include the arc zero and the arc of ninety degrees. Of its 
functions, the limits of the sine, are zero and radius ; zero for 
the arc zero, and radius for the arc of ninety degrees. For 
the tangents, the limits are zero and infinity ; zero for the 
arc zero, and infinity for the arc of ninety degrees ; and 
similarly for all the other functions. For all numbers, the 
limits are zero and infinity ; and for all algebraic quantities, 
minus infinity and plus infinity. 

When we consider continuous quantity, we find the 
second definition in direct conflict with the first Lemma 
of Newton, which has been well called, " the corner- 



36 INTRODUCTION" TO 

stone and foundation of the Principia" It is very 
difficult to comprehend that two quantities may ap- 
proach each other in value, and in any given time become 
Rearer equal than any given quantity, and yet never become 
equal ; not even when the approach can be continued to 
infinity, and when the law of change imposes no limit to 
the decrease of their difference. This, certainly, is contrary 
to the theory of Newton. 

Take, for example, the tangent line to a curve, at a given 
point, and through the point of tan gen cy draw any secant, 
intersecting the curve, in a second point. If now, the 
second point be made to approach the point of tangency, 
both definitions recognize the angle which the tangent line 
makes with the axis of abscissas as the limit of the angles 
which the secants make with the same axis, as the second 
point of secancy approaches the tangent point. By the first 
definition, the supposition of consecutive points causes the 
secant line to coincide with, and become the tangent. But 
by the second definition, the secant line can never become 
the tangent, though it may approach to it as near as we 
please. This is in contradiction to all the analytical 
methods of determining the equations of tangent lines to 
curves. See corollaries 1, 2, 3, and 4 of lemma III. ; in which 
all the quantities referred to are supposed to reach their 
limits. 

By the second definition, there would seem to be an im- 
passable barrier placed between a variable quantity and its 
limit. If these two quantities are thus to be forever sepa- 
rated, how can they be brought under the dominion of a 
common law, and enter together into the same equation. 
And if they cannot, how can any property of the one be 



DIFFEBENTIAL CALOULUS. 3? 

Used to establish a property of the other ? The mere fact 
of approach, though infinitely near, would not seem to 
furnish the necessary conditions. 

The difficulty of treating the subject in this way is strik- 
ingly manifested in the supplementary definition of an infi- 
tesimal. It is defined, simply, as " every variable magnitude 
whose limit is zero/ 9 

Now, may not zero be a limit of eyery variable which has 
not a special law of change ? Is not this definition too 
general to give a definite idea of the individual thing de- 
fined — an infinitisimal ? We have no crystallized notions of 
a class, till we apprehend, distinctly, the individuals of the 
class — their marked characteristics — their harmonies and 
their differences ; and also, their laws of relation and con- 
nection. 

Having given and illustrated these definitions, M. Duha- 
mel explains the methods by which we can pass from the 
infinitesimals to their limits ; and when, and under what 
circumstances, those limits may be substituted and used for 
the quantities themselves. Those methods have not seemed 
to me as clear and practical as those of Newton and Leib- 
nitz. 

It is essential to the unity of mathematical science, that 
all the definitions, should, as far as possible, harmonize with 
each other. In all discontinuous quantities, the boundaries 
are included, and are the proper limits. In the hyperbola, 
for example, we say that the asymptote is the limit of all 
tangent lines to the curve. But the asymptote is the tan- 
gent, when the point of contact is at an infinite distance 
from the vertex : and any tangent will become the asymp 
tote, under that hypothesis. 



38 INTRODUCTION TO 

If s denotes any portion of a plane surface, y, the ordinate 
and x the abscissa, we have the known formula : 

ds = ydx. 

If we integrate between the limits of x = 0, and x — a % 
we have, by the language of the Calculus 



r =/ 



% y dx, 



which is read, "integral of the surface between limits 
of x = 0, and x = a" in which both boundaries enter into 
the result. 

The area, actually obtained, begins where x = 0, and ter- 
minates where x = a, and not at values infinitely near those 
limits. 

What Quantities are denoted by 0. 

23. Our acquaintance with the character 0, begins in 
Arithmetic, where it is used as a necessary element of the 
arithmetical language, and where it is entirely without value, 
meaning, absolutely nothing. Used in this sense, the 
largest finite number multiplied by it, gives a product equal 
to zero ; and the smallest finite number divided by it, gives 
a quotient of infinity. 

* When we come to consider variable and continuous quan- 
tity, the infinitesimal or element of change from one con- 
secutive value to another, is not the zero of Arithmetic, 
though it is smaller than any number which can be ex- 
pressed in terms of one, the base of the arithmetical system 



DIFFERENTIAL CALCULUS. 39 

Hence, the necessity of a new language. If the variable is 
denoted by z, we express the infinitesimal by dx; if by y, 
then hjdy; and similarly, for other variables. 

Now, the expressions dx and dy, have no exact synonyms 
in the language of numbers. As compared with the unit 1, 
neither of them can be expressed by the smallest finite part 
of it. Hence, when it becomes necessary to express such 
quantities in the language of number, they can be denoted 
only by 0. Therefore, this 0, besides its first function in 
Arithmetic, where it is an element of language, and where 
the value it denotes is absolutely nothing, is used, also, to 
denote the numerical values of the infinitesimals. Hence, 
it is correctly defined as a character which sometimes 
denotes absolutely nothing, and sometimes an infinitely 
small quantity. We now see, clearly, what appears obscure 
in Elementary Algebra, that the quotient of zero divi- 
ded by zero, may be zero, a finite quantity, or infinity. 

Inscribed and circumscribed Polygons unite on the Circle. 

•24. The theory of limits, developed by Newton, is not 
only the foundation of the higher mathematics, but indi- 
cates the methods of using the Infinitesimal Calculus in the 
elementary branches. This Calculus being unknown to the 
ancients, their Geometry was encumbered by the tedious me- 
thods of the reductio ad absurdum. Newton says in the scho- 
lium : " These lemmas are premised to avoid the tedious- 
ness of deducing perplexed demonstrations ad absurdum, 
according to the method of the ancient geometers." 

Lemma L, which is the " corner-stone and foundation of 
the Principia" is also the golden link which connects ge- 
ometry with the higher mathematics. 



40 INTRODUCTIOK TO 

It is demonstrated in Euclid's Elements, and also in Da- 
vies' Leqendre, Book V., Proposition X., that " Two regular 
polygons of the same number of sides can be constructed, the 
one circumscribed about the circle and the other inscribed 
within it, which shall differ from each other by less than any 
given surface" 

The moment it is proved that the exterior and interior 
polygons may be made to differ from each other by less 
than any given surface, Lemma I. steps in and affirms an ulti- 
mate equality between them. And when does that ultimate 
equality take place, and when and where do they become 
coincident ? Newton, in substance affirms, in his lemmas, 
" on their common limit, the circle," and under the same 
hypothesis as causes the inscribed and circumscribed paral- 
lelograms to become equal to their common limits, the 
curvilinear area. If Lemma I. is true, the perimeters of the 
twp polygons will ultimately coincide on the circumference 
of the circle, and become equal to it. But what then is the 
side of each polygon ? We answer, the distance between 
two consecutive points of the circumference of the circle? 
And what is that value ? We answer, the -yj dx* + dy*. 

But it is objected, that this introduces us to the infinitely 
small. True, it does; but we cannot reach a continuous 
quantity without it. The sides of the polygons, so long as 
their number is finite, will be straight lines, each diminishing 
in value as their number is increased. While this is so, the 
perimeter of each will be a discontinuous quantity, made up 
of the equal sides, each having a finite value, and each being 
the unit of change, as we go around the perimeter. As each of 
these sides is diminished in value, and their number increas- 
ed, the discontinuous quantity approaches the law of conti- 



DIFFEKENTIAL CALCULUS. 41 

nuity, which it reaches, under the hypothesis, that each 
side becomes infinitely small and their number infinitely 
great. Behold the polygons embracing each other on their 
common limit, the circle, and the perimeter of each coinci- 
ding with the circumference. Thus, the principles of the In- 
finitesimal Calculus take their appropriate place in Elemen- 
tary Geometry, to the exclusion of the cumbrous methods of 
the reductio ad dbsurdum of the ancients, and the whole 
science of Mathematics is brought into closer harmonies 
and nearer relations. 

Differential and Integral Calculus. 

25. We have seen that the Differential and Integral Cal- 
culus is conversant about continuous quantity. We have 
also seen, that such quantities are developed by considering 
their laws of change. We have further seen, that these 
laws of change are traced by means of the differences of 
consecutive values, taken two and two, as the variables pass 
from one state of value to another. Indeed, those differ- 
ences are but the foot-steps of these laws. 

Language of the Calculus. 

26. We are now to explain the language by which the 
quantities are represented, by which their changes are indi- 
cated, and by which their laws of change are traced. 
The constant quantities which enter into the Calculus are 
represented by the first letters of the alphabet, a, 5, c, etc., 
and the variables, by the final letters, x, y, z, etc. 

When two variable quantities, y and #, are connected in 
an equation, either of them may be supposed to increase or 
decrease uniformly ; such variable is called the independent 

2* 



42 l^TKODUCTION TO 

variable, because the lata of change is arbitrary, and inde- 
pcndc?it of the form of the equation. This variable is gener- 
ally denoted by x, and called simply, the variable. Under 
tiiis hypothesis, the change in the variable y will depend on 
the form of the equation : hence, y is called the dependent 
variable, or function. When such relations exist between y 
and x, they are expressed by an equation of the form 

y = F (x), y = f (x), or, / (x, y) = 0, 

which is read, y a function of x. The letter F, or /, is a 
mere symbol, and stands for the word function. If y is a 
function of x, that is, changes with it, x may, if we please, 
be regarded as a function of y ; hence, 

One quantity is a function of another, when the ttvo are so 
connected that any change of value, in either, produces a cor- 
responding change in the other. 

It has been already stated (Art. 8), that the difference 
between two consecutive values of a variable quantity, is 
indicated by simply writing the letter d as a symbol, before 
the letter denoting that variable ; so that dx denotes the 
difference between two consecutive values of the variable 
quantity denoted by x, and dy the difference between the 
corresponding consecutive values of the variable quantity 
denoted by y. These are mere forms of language, express- 
ing laws of change. 

How are the changes in these variable quantities, ex- 
pressed by the infinitesimals, to be measured ? Only by tak- 
ing one of them as a standard — and finding how many times 
it is contained in the other. 

The independent variable is always supposed to increase 



DIFFERENTIAL CALCULUS. 43 

uniformly ; hence, the difference between any two of its 
consecutive values, taken at pleasure, is the same : therefore, 
this difference, which does not vary in the same equation, 
or under the same law of change, affords a convenient 
standard, or unit of measure, and in the Calculus, is always 
used as such. 

The change in the function y, denoted by dy, is always 
compared with the corresponding change of the independent 
variable, denoted by dx, as a standard, or unit of measure. But 
the change in any quantity, divided by the unit of measure, 

gives the rate of change : hence, ~ is the rate of change of 

the function y. This rate of change is called the differ- 
ential coefficient of y regarded as a function of x, and per- 
forms a very important part in the Calculus. The quanti- 
ties dy and dx, being both infinitesimals, are of the same 
species : hence, their quotient is an abstract number. There- 
fore, the differential coefficient is a connecting link between 
the infinitesimals and numbers. 

If any quantity whatever be divided by its unit of meas- 
ure, the quotient will be an abstract number; and if this 
quotient be multiplied by the unit of measure, the product 
will be the concrete quantity itself. Hence, if we multiply 

-—, by the unit of measure dx, we have ~ dx, which always 

denotes the difference between two consecutive values of y ; 
and therefore, is the differential of y. Hence, the differen- 
tial of a variable function is equal to the differential coeffi- 
cient multiplied by the differential of the independent variable. 

The method, therefore, of dealing with infinitesimals, is 



44 INTEODUCTION, ETC. 

precisely the same as that employed for discontinuous 
quantities. 

We assume a unit of measure which is as arbitrary as one, 
in numbers, or, as the foot, yard, or rod, in linear measure, 
and then we compare all other infinitesimals with this 
standard. We thus obtain a ratio which is an abstract 
number, and if this be multiplied by the unit of measure, 
we go back to the concrete quantity from which the ratio 
was derived. 

We have thus sketched an outline of the Infinitesimal 
Calculus. We have named the quantities about which it is 
conversant, the laws which govern their changes of value, 
and the language by which these laws are expressed. We 
have found here, as in the other branches of mathematics, 
that an arbitrary quantity, assumed as a unit of measure, 
is the base of the entire system ; and that the system itself 
is made up of the various processes employed in finding the 
ratio of this standard, to the quantities which it measures. 



DIFFERENTIAL CALCULUS. 



SECTION 1. 

DEFINITIONS AND FIRST PRINCIPLES. 
Definitions. 

1. In the Differential Calculus, as well as in Analyt- 
ical Geometry, the quantities considered are divided intc 
two classes: 

1st. Constant quantities, which preserve the same values 
m the same investigation ; and, 

2d. Variable quantities, which assume all possible values 
that will satisfy any equation vihich expresses the relation 
between them. 

The constants are denoted by the first letters of the 
alphabet, a, J, c, &c. ; and the variables, by the final let- 
ters, cc, 2/, z, &c. 

Uniform and varying changes. 

2. There are two ways in which a variable quantity 
may pass from one value to another. 

If the variable <e, once had the particular value, x = a, 
and afterwards assumed the value, x ±= a\ we can sup 
pose : 



46 DIFFERENTIAL CALCULUS. [SEC. L 

1st. That during the change from a to a\ x assumed, 
in succession, and by a uniform change, all the values 
between a and a\ just as a body moving uniformly over 
a given straight line passes through all the points between 
its extremities; or, 

2d. We may suppose, that during the change from a to 
a\ x assumed all possible values between its limits, with- 
out the condition of a uniform change. In both cases, 
the quantity is said to be continuous. 

3. If two variable quantities, y and &, are connected 
in an equation, as, for example, 

y = & + 2 ; 

then, to every value of cc, arbitrarily assumed, there will 
be a corresponding value of y, dependent upon, and result- 
ing from, the value attributed to x. Thus, if we make 
x = 4, we have, . 

y = 16 + 2 = 18. 

If we suppose x to increase from 4 to 5, we shall have, 

y = 25 -h 2 = 27; 

thus, while x changes from 4 to 5, y changes from 18 to 27. 
If now we suppose x to increase from 5 to 6, y will 
increase from 27 to 38. Thus, while x increases uniformly 
by 1, y will change its value according to a very different 
law. 

Function and variable. 

4. When two variable quantities, y and x, are con- 
nected in an equation ; either of them may be supposed 



8E0. I.] FIRST PRINCIPLES, 47 

to increase or decrease uniformly ; a variable, so changing, 
is called the independent variable, because the law oj 
change is arbitrary^ and independent of the form of equa- 
tion. This variable is generally denoted by x, and called 
simply, the variable. The change in the variable y, de- 
pends on the form of the equation ; hence, y is called 
the dependent variable, or function. When such a rela- 
tion exists between y and x, it is expressed by an equa- 
tion of the form, 

y = F(x), y = f(x) ; or, f(y, x) = ; 

which is read, y a function of x. The letter F, or /, 
is a mere symbol, and stands for the w r ord, /miction. If 
y is a function of x, that is, changes with it, x is also 
a function of y\ hence, 

One quantity is a function of another, ichen the two 
are so connected that any change of value, in either, pro- 
duces a corresponding change in the other. 

5. If the equation connecting y and x, is of such a 
form that y occurs alone, in the first member, y is called 
an explicit function of x. Thus, in the equations, 

y = ax + b of a straight line, 

y = <\fl& — x 2 .... of the circle, 

y — -- ^/A 2 — x 2 ... of the ellipse, 

y = -\/2px . . . . . of the parabola, and 
B 



y — —■ ^/x 2 — A 2 ... of the hyperbola, 
y is an explicit function of x. 



CALCULUS. [SEC 


;ten under the forms, 


or, 


Vtay) = o, 


or, 


f{*,y) = o, 


or, 


f{x,y) = 0, 


or, 


/(*, y) = o, 


or, 


/(*, y) = o, 



48 DIFFERENTIAL 



y — ax — b = 0; 

y 2 + x 2 - R 2 = 0; 

^L 2 y 2 + .£ 2 a 2 - ^L 2 jB 2 = ; 

y 2 — 2px == ; 

^2^2 _ ^2 + ^2JJ2 _ ; 

y is called an implicit function of x ; the nature of the 
relation between y and x being implied, but not developed 
in the equation. 

6. It is plain, that in each of the above equations 
the absolute value of y, for any given value of #, will 
depend on the constants which enter into the equation* 
this relation is expressed, by calling y an arbitrary function 
of the constants on which it depends. Thus, in the equa- 
tion of the straight line, y is an arbitrary function of a 
and b ; in the equation of the circle, y is an arbitrary 
function of R ; in the equation of the ellipse, of A and 
B ; in the equation of the parabola, of 2p ; and in the 
equation of the hyperbola, of A and B. 

7. An increasing function is one which increases when 
the variable increases, and decreases when the variable 
decreases. A decreasing function is one which decreases 
when the variable increases, and increases when the variable 
decreases. 

In the equation of a straight line, in which a is posi- 
tive, y is an increasing function of x. In the equations 
of the circle and ellipse, y is a decreasing function of x. 
In the equation of the parabola, y is an increasing func- 
tion of x. In the equation of the hyperbola, y is imaginary 



SEO. I.] FIRST PKINCIPLR8, 40 

for all values of x < A, and an increasing function for 
all positive values of x > A. 

N. A quantity may be a function of two or more vari- 
ables, If 

U = ax + by 2 , or u = ax 2 — by 2 + cz + d, 

u will be a function of x and y, in the first equation, 
and of x, y, and 2?, in the second. These expressions may 
be thus written : 

^ = /(«> y)> and w = /(^ y> s )- 

If, in the second equation, we make, in succession, the 
independent variables cc, y, and 3, respectively equal to 
0, Ave have, 

for, <c=0, u— —by 3 + cz+d =/(y, 3), 

for, x—0, and y = 0, zj = C3 + c? =f(z)\ and, 

for, 05=0, y=0, and 3 = 0, u = d =a constant. 

Algebraic and Transcendental Functions. 

9, There are two general classes of functions : 

Algebraic and Transcendental. 

Algebraic functions are those in which the relation be 
tween the function and the variable can be expressed in 
the language of Algebra alone : that is, by addition, sub- 
traction, multiplication, division, the formation of powers 
denoted by constant exponents, and the extraction of roots 
indicated by constant indices. 

Transcendental functions are those in which the relation 
between the function and variable cannot be expressed ir 
the language of Algebra alone. There are three kinds : 



50 



DIFFERENTIAL CALCULUS. 



[SEC. I. 



1. Exponential functions, in which the variable enters as 

an exponent ; as, 

u = a x . 

2. Logarithmic functions, which involve the logarithm 

of the variable ; as, 

u ~ logo;. 

3. Circular functions, which involve the arc of a circle, 
or some function of the arc; as, 



u = sin a, 



u = cosa, 



u = tana. 



Geometrical representation of Functions. 

10. With the aid of Analytical Geometry, it is easy to 
trace, geometrically, the numerical relation between any 
function and its independent variable. 

Suppose we have given the equation, 

V = /(*)• 

If we attribute to #, the independent variable, in succes- 
sion, every value between — oo and + oo, each will give 
a corresponding value for y, which may be determined 
from the equation, y = f(x) 

Let be the origin of a 
system of rectangular co-or- 
dinates. From 0, lay off to 
the right, all the positive 
values of cc, and to the left all 
the negative values. Through 
the extremity of each abscissa, 
so determined, draw a line 
parallel to the axis of ordinates, 
and equal to the corresponding value of y\ the plus values 




SEC. I.j 



FIRST PRINCIPLES. 



51 



will fall above the axis of JT, and the negative values below 
it; then trace a curve, AMN, through the extremities of 
these ordinates. The co-ordinates of this curve will indicate 
every relation between y and cc, expressed by the equation. 

V = /(«). 

This curve should present to the mind, not merely any 
particular value of #, and the corresponding value of y, 
but the entire series of corresponding values of these two 
variables. 

Quantities infinitely small — Differentials. 

II. A quantity is infinitely s?nall, when it cannot bs 
diminished, according to the law 
of change, without becoming 0. 

If, in the equation of the curve, 

y =/(«) . . (1.) 

x ha* a particular value OP, y 
will denote the ordinate PM. 

If x be increased by JPQ, de- 
noted by A, PifcTwill change to 
QN, which we will denote by y' ; and we shall have, 

y' = f(x + h) (2.) 

If we subtract equation (1) from (2), we obtain, 

y'-y=f( x + h)-f{x) .... (3.) 

It is evident that each member of this equation will reduce 
to 0, when we make h = 0. 




52 



DIFFERENTIAL CALCULUS. 




Let us suppose, as before, that the abscis x has increased from 
OP to OQ, and that the cor- 
responding ordinate y, has be- 
come y f . Draw through the 
points JV and M, the secant 
line iOT. If, now, we suppose 
the point JV to approach M, 
till it becomes consecutive with 
it, then, 

1. The secant line will become the tangent SMT\ 

2. The abscissas OP and OQ will become consecutive; 

3. The ordinates Pi!/", QJST, will also be consecutive. 

The differential of a quantity is the difference between 
any two of its consecutive values; hence, it is indefinitely small. 

The differential is expressed, by writing d before the 
letter denoting the quantity : thus, dx denotes the differen- 
tial of #, and is read, differential of x : dy denotes the 
differential of y, and is read, differential of y. 

It is plain, that dx denotes the last value of A, in Equa- 
tion (3), before it becomes 0; and ^that dy denotes the last 
difference between y' and y, as A approaches to 0. 



Differential Coefficient. 

12. Under the preceding hypotheses, the differentials of 
x and y admit of geometrical interpretations. 

If we divide both members of Equation (3) by A, we have, 

y'-y = A* + h ) -A") 

h h 



(4) 



Having drawn MR parallel to the axis of abscissas, NR 
will denote the difference of the two ordinates y f and y\ 
hence, the first, and consequently the second member of 
Equation (4), will denote the tangent of the angle NMR^ 



SEC. r.] FIK ST PRINCIPLES. 53 

which the secant line makes with the axis of -5T. Denote 
the angle which the tangent makes with the axis of X by a. 
When the ordinates y' and y become consecutive, the 
secant NM becomes tangent to the curve at the point M, 
and the angle NMR becomes equal to TSP \ and we have, 

^ = tan a (5.) 

dx 

The term —-, is called the differential coefficient of y, 

regarded as a fraction of x ; hence, 

The differential coefficient of a function is the differen- 
tial of the function divided by the differential of the 
independent variable. 

If we multiply both members of Equation (5) by dx, 

^-dx = tan a dx\ 
dx 

but the tan a multiplied by the base dx, of the indefinitely 
small triangle, is equal to the perpendicular, which is the 
difference between the consecutive values of y' and y, de 
noted by dy\ therefore, 

-^■dx = dy ; hence, 
dx 

The differential of a function is equal to its differential 
coefficient multiplied by the differential of the variable. 



Limiting Ratio. 
13. Let us now resume the consideration of Equation (4). 

y' - y _ ffr + h)-M u) 

h ~~ h • • • v •; 

The first member of this equation is the ratio of the 
increment A, of the independent variable, to the correspond 



54 DIFFERENTIAL CALCULUS. [sEC. I. 

ing increment of the function, and denotes the tangent of 
the angle which the secant line, drawn through the ex- 
tremities of y and y\ makes with the axis of abscissas. 

If we suppose h to decrease, the secant line will ap- 
proach the tangent, and the ratio will approach the tangent 
of the angle which the tangent line makes witli the axis oi 
X. The tan a, is, therefore, the limit of this ratio, and 
since it is also the differential coefficient, it follows that, 

The differential coefficient is the limit of the ratio of the 
increment of the independent variable to the increment of 
the function, 

A varying ratio, of any increment of the independent 
variable denoted by A, to the corresponding increment of 
the function, denoted by y' — y, reaches its limit when h 
reaches its last value; and then, the values of y' and y 
become consecutive; therefore, the limiting ratio, is the 
ratio of consecutive values. Hence, if w r e have an expres- 
sion for the ratio of the increments, we pass to the limiting 
ratio, or differential coefficient, by making h indefinitely 
small. 

Form of the difference between two states of a function. 
14. Let us resume the discussion of Equation (3). 

y>-.y=f X + h)-jXx). 

If h be made equal to 0, the first and second members 
will each reduce to 0. Therefore, if the second member be 
developed, and the like terms having contrary signs can- 
celled, each of the remaining terms will contain h ; else, all 
the terms would not reduce to 0, when h = 0. Hence, the 
second member of Equation (3) is divisible by A. Dividing 
by A, we have, 

y'-V _ A x + h )~f x . . . (4.) 



RKC. I.] FIB ST PRINCIPLES. 55 

If, now, we pass to the limiting ratio, by making h in- 
definitely small, the second member will become the tan a, 
a quantity independent of h (see Equation 5); hence, the 
first term in the development of the second member of 
Equation (3) contains h only in the first power, and the 
coefficient of this term is tan a, or the differential co- 
efficient. Since all the other terms become 0, when h = 0, 
each of them must contain A to a higher power than the 
first. 

If we designate by .P, the differential coefficient of y, and 
by P' such a value that P'h? shall be equal to all the 
terms of the development of the second member of Equation 
(3), after the first, that equation may be written under the 
form 

y'-y = Ph + P'h* .... (6.) 

The differential coefficient, P, is independent of A, but 
will, in general, contain x\ and when it does, it is a func- 
tion of that variable : P\ when not equal to 0, is a function 
of x and h. 

Applications of the Formula. 

1. If we have an expression of the form, 

V =f( x ) = ax , 
we have the form or development of the second mem 
ber. 

If we give to x an increment A, we have, 

y' =z A x + ^) — a ( x + *) = ax + ah\ hence, 
y' — y —f{x + h) —f(x) = ah; and 

^ — — ^ = a ; passing to consecutive values, 
h 

Jf- = a; and -Idx == adx. 
ax dx 



56 DIFFERENTIAL CALCULUS. [SEC. I. 

2. If we have a function of the form, 

V = f( x ) =ax z , 

we again have the form or development of the second 
member. 

y' =/(« + h) = a(x + h)* = ax 3 + 3ax 2 h + 3axh 2 + ah 3 
y' - y = f(x + h) -f(x) = 3ax 2 h + 3aa^ 2 + ah 3 

y ~ y = 3ax 2 + 3axh + ah 2 ; 

passing to consecutive values, we have, 

% = 3ax 2 ; and ^dx = 8aa5 a <fc. 

In the first example P = a, and P' = 0. In the 
second, P = 3ax 2 , and P' = 3axh + ah 2 . 

15. Equation (6) affords the means of determining the 
differential coefficient, and the differential of any function, 
whose form is developed in terms of the independent 
variable. 

If we divide both members of Equation (6) by the in- 
crement A, and then pass to the limiting ratio, we have the 
differential coefficient. If we then multiply the differential 
coefficient by the differential of the independent variable, 
we have the differential of the function. 

Equal functions have equal differentials 

16. If two functions, u and w, dependent on the same 
variable as, are equal to each other, for all possible values 
of #, their differentials will also be equal. 



SEC. I.] FIRST PRINCIPLES. 57 

For, x being the independent variable, we have (Art. 14), 
u > _ u = Ph + P'h\ 

v' — v = Qh+ Q'h\ 

in which P is the differential coefficient of u, regarded 
as a function of #, and Q the differential coefficient of 
v, regarded as a function of cc. 

But, since u' and v f are, by hypothesis, equal to each 
other, as well as u and v, we have, 

Ph + P f h? = Qh + Q'h\ 

or, by dividing by h and passing to consecutive values, 

P = Q, 

, du dv 

dx ~~ dx' 

t du 7 dv ^ 

and, — ccx = — dx, 

that is, the differential of w is equal to the differential 
of v. 

Converse not true. 

17. The converse of this proposition is not generally 
true ; that is, 

If two differentials are equal to each other, ice are not 
at liberty to conclude that the functions from which they 
xcere derived, are also equal. 

For, let u — v ± A (1.) 

in which A is a constant, and u and v both functions 
of x. Giving to x an increment A, we shall nave, 

u f = v f ±l A, 



58 DIFFERENTIAL CALCULUS. [SEC. I. 

from which subtract Equation ( 1 ), and we obtain, 

u' — u = v' — w, 
and, by substituting for the difference between the two 
states of the function, we have, 

Ph + Fh? = Qh 4- Q'h\ 

Dividing by A, and passing to consecutive values^ we 

obtain, 

-~ „ , . du dv 

P= Q; that is, Tx = ^, 

du 7 dv ^ n , 

hence, — dx ~ — dx : or, du = a# . 

a# ace 

Hence, the differentials of ££ and v are equal to each other, 
although v may be greater or less than u, by any constant 
quantity A ; therefore, 

Every constant quantity connected with a variable by 
the sign plus or minus, will disappear in the differen- 
tiation. 

The reason of this is apparent ; for, a constant does not 
increase or decrease with the variable ; hence, there is no 
ultimate or last difference between two of its values; and 
this ultimate or last difference is the differential of a 
variable function. Hence, the differential of a constant 
quantity is equal to 0. 

18. If we have a function of the form, 

u = Av, 

in which u and v are both functions of x, and give to 
x an increment A, we shall have, 

u' — u = A(v' — #), 
or, Ph + Fh* = A(Qh + Q f h% 



SEC. I.J FIRST PRINCIPLES. 59 

Dividing by A, and passing to the consecutive values, 

P= AQ, 

or, Pdx == AQdx. 

But, du = Pdx, and efo = Qdx ; 

hence, $w = Adv\ that is, 

7%6 differential of the product of a constant by a 
variable quantity, is equal to the constant multiplied by 
the differential of the variable. 

Signs of the differential coefficient, 

19. If u is any function of x, and we give to x a& 
increment A, we have, 

^ = P + rh . 

and since h is positive, the sign of the first member wil 
be positive when u < u r ; that is, when ^ is an incream% 
function of x (Art. 7). It will be negative when u > u' \ 
that is, when m is a decreasing function of x. Passing 
to consecutive values, we have, under the first supposition, 

— = + P: and 

— = — P, under the second; hence* 

7%6 differential coefficients have the same sign as the 
functions, when the functions are increasing, and con- 
trary signs, when they are decreasing. 

If we multiply by dx, we obtain the differentials, which 
have the same signs as the differential coefficients. 



GO DIFFERENTIAL CALCULUS. [SEC. I 

Nature of a differential coefficient, and of a differential. 

20. The method of treating the Differential Calculus, 
adopted in this treatise, is based on three hypotheses: 

1st. That the independent variable changes uniformly : 
2d. That in changing from one state of value to 
another, it passes through all the intermediate values; and, 
3d. That any function dependent upon it, undergoes 
changes determined by the equation expressing the rela- 
tions between them ; and that such equation preserves the 
same general form. 

If the independent variable changes uniformly, and as- 
sumes all possible values between the limits x = a, and 
x = a', we have seen that the change cannot be denoted 
by a number. If, then, we denote this change by dx, we 
mean that dx is smaller than any number ; hence, 

dx 

r% * i ax ax ax p 

But, — - = — - = — — = > <fcc. 

dx dx 2 dx 3 dx* 

that is, any power of dx divided by a power of dx 
greater by 1, is infinite; hence, any power of dx is infin 
itely small, compared with the power next less. Hence, it 
follows : 

1st. That the addition of dx to any number, can make 
no alteration in its value ; and therefore, when connected 
with a numeral quantity by the sign ± , may be omitted 
without error ; thus, 

Sax + dx = Sax. 



SEC. I.] FJRST PRINCIPLES. CI 

2d. Since dx 2 ie infinitely small, compared withe&c; that 
is, infinitely less than dx, we have, 

5ax 2 dx + dx 2 = 5ax 2 dx-, 

and similarly for the higher powers of <fe. 

The quantities, e&c, dx 2 , dx 3 , &c, are called infinitely 
small quantities, or infinitesimals of the first, second, and 
third orders : from their law of formation, it follows that, 

Every infinitely small quantity may he omitted without 
error when connected by the sign ± with any of a lower 
order. 

Rate of change. 

21. The measure of a quantity, great or small, is the 
number of times which it contains some other quantity 
of the same kind, regarded as a unit of measure. 

In the Differential Calculus, dx, the differential of the 
independent variable, is the unit of measure. The rate of 

du 
change, in the function y, is therefore expressed by ~ , 

and the actual change corresponding to dx, by 

% dx = *»- 

22. The equation of a straight line is, 

y — ax + b. 

If we take any point, as M, whose co-ordinates are y 
and x, and a second point JV, whose co-ordinates are y', 
x + A, and we have, 

y' - y = ah; or, y y = a . . (1.) 



62 
that is, 



DIFFERENTIAL CALCULUS. 



[SEC. L 



MB 



= tangent NMR = a ; 



and, passing to the consecutive 

values, 

dy 




—■ = tangent a = a . 
dy 



(2.) 



The differential coefficient -~ , measures the rate of 

increase of the ordinate y, when x receives the incre- 
ment dx; and since this value is independent of #, the 
rate will be the same for every point of the line ; that 
is, the rate of ascension of the line from the axis of 
abscissas, is the same at every point. And since, 

~- dx = dy = adx, 
ax 

the change in the value of the ordinate will be uniform, 
for uniform changes in the abscissa. 

23. Let us examine an 
equation, 

y =/(x) . . (1.) 

not of the first degree. 

Let us suppose the curve 
AMN to be such that the 
abscissas and ordinates of its 
different points shall correspond 
to all possible relations between y and x, in Equation ( 1 ). 

We have seen (Art. 13) that, 




* == tan TSP 
ax 



tan a ; hence, 



SEC, I.] FIRST PRINCIPLES. 63 

the rate of increase of the function, or the ascension of 
the curve at any point, is equal to the tangent of the 
angle which the tangent line makes with the axis of ab- 
scissas. We also see, that this value of the tangent of 
a, will vary- with the position of the point M; hence it 
is a function of x\ therefore, 

In every equation, not of the first degree, the differ 
ential coefficient is a function of the independent variable, 

1 We have seen, that when the points M and N are 
consecutive, the secant line, MN, becomes the tangont 
iine, TMS (Art. 13). The line MR is then denoted by 
dx, and RN~ or RT, (for the points N and T then coin- 
cide), by dy. If we give to the new abscissa, x + dx, 
an additional increment dx, and suppose the correspond- 
ing ordinate, y + dy, to receive the same increment as 
before, viz. : dy, the extremity of the last ordinate will 
not fall on the curve, but on the tangent line, since the 
triangles thus formed are similar ; hence, 

If a function be supposed to increase uniformly from 
any assumed value, the differential coefficient will be 
constant, and equal to any increment of the function 
divided by the corresponding increment of the variable. 

Nature of the Differential Calculus. 

24. In every operation of the Differential Calculus, one 
of two things is always proposed, and sometimes both : 



64 DIFFERENTIAL CALCULUS. [SEC. I 

1st. To find the rate of change in any variable func- 
tion when it begins to change from any assigned value. 

2d. To find the difference between any two consecutive 
values of the function. This difference is the actual 
change in the function, produced by the smallest change 
which takes place in the independent variable. 

The use of the independent variable is to furnish a unit 
of measure for the increment of the function, and thus to 
determine its rate of change, as it passes through all its 
states of value. This ratio can generally be expressed in 
numbers, either exactly or approximatively. 

25. The increment of the function, corresponding to 
the smallest increment of the variable, being the difference 
between any two of its consecutive values, is a quantity of 
the same kind as the function, and differs from it only in 
this : that it is too small to be expressed by numbers. The 
differential of a quantity, therefore, is merely an element 
of that quantity ; that is, it is the change which takes place 
when the quantity begins to increase or decrease, from 
any assumed value. When we find this element, we have 
the differential of the function ; and by dividing by dx^ 
we have the differential coefficient. Hence, 

All the operations of the Differential Calculus comprise 
but two objects : 

1. To find the rate of change in a function, when it 
passes from one state of value to another, consecutive 
with it. 

2. To find the actual change in the function. 

The rate of change is the differential coefficient, and the 
actual change, the differential. 



SECTION II. 

DIFFERENTIALS OF ALGEBRAIC FUNCTIONS 
Differential of sum or difference of Functions* 

26. Let as be a function of the algebraic sum of 
several variable quantities, of the form, 

u=y+z-w= f(x), 

in which y, 2, and w, are functions of the independent 
variable x. 

If we give to x an increment A, we shall have, 

u' — u = (y f — y) + (z' — z) — (w r — w) ; 

hence (Art. 14), 

u' -u = (Ph + P'A 2 ) + ( Qh + Q'h?) - (Lh + L f h% 

or, U —=^ = (P + P'A) + (C + Q'A) - (Z + Z'A), 

and by passing to consecutive values, 

multiplying both members by c?cc, we have, 

— dx =1 Pdx + Qdx — Zdx. 
dx 

But as P, §, and i, are the differential coefficient? 



66 DIFFERENTIAL CALCULUS. [SEC. II. 

of y, z, and w, each regarded as a function of x\ hence, 

** = ** + **- **i that is, 

ax ax ax ax 

The differential of the sum or difference of any number 
of functions, dependent on the same variable, is equal to 
the sum or difference of their differentials taken separately 

Differential of a product. 

2*7. Let u and v denote any two functions, x the 
independent variable, and h its increment ; we shall then 
have, 

u' =: u + Ph + P'h 2 , and 

v' = v + Qh + Q'h\ 

and, multiplying, 

u'v' = (u + Ph+ P'A 2 ) (v + Qh + Q'h*) 
= uv + vPh + uQh + PQh 2 + &c. ; 
hence, 

, = vP + u Q + terms containing A, A 2 , and A 3 . 

If now we pass to consecutive values, we have, 

therefore, d(uv) = vPdx + u Qdx = vdu + udv ; hence, 

7%^ differential of the product of two functions de- 
pendent on the same variable, is equal to the sum of 
the products obtained by multiplying each by the differ- 
ential of the other. 



SEC, 11. J DIFFEEEKTIALS OF FUNCTIONS. 67 

1. II* we divide by uv, we have, 

d(uv) du dv . . 
= 1 \ w 

uv u v 

that is, 

Tlie differential of the product of two functions, divided 
by the product, is equal to the sum of the quotients 
obtained by dividing the differential of each by its 
function. 

2§. We can easily determine, from the last formula, 
the differential of the product of any number of functions. 
For, put v — ts, then, 

dv d(ts) dt ds ■ 

H = ~is~ = 7 + T ' ' ' ' (2 ' } 

and by substituting ts for v, in Equation ( 1 ), we have. 

d(uts) du dt ds 
uts u t s * 

and in a similar manner we should find, 

diutsr. . . .) du dt ds , dr « 

— = 1 1 1 .... &c. 

utsr .... u t s r 

If, in the equation, 

d(uts) du dt ds 
uts uts 

we multiply by the denominator of the first member, we 
shall have, 

d(uts) = tsdu + usdt + utds; hence, 

The differential of the product of any number of func 
tions, is equal to the sum of the products which arise 



68 DIFFERENTIAL CALCULUS. [SEC. II. 

by multiplying the differential of each function by the 
product of all the others. 



Differentials of Fractions. 

29. To obtain the differential of any fraction of the 

,. u 

form, - • 

v 

u 

Put, - = L then, u = tv. 

v 

Differentiating both members, we have, 
du es vdt + tdv; 

finding the value of dt, and substituting for t its value 

- , we obtain, 

v 

, du udv 
dt — , 

V V 2 

or, by reducing to a common denominator, 

vdu — udv , 
dt = - ; hence, 

The differential of a fraction is equal to the denom- 
inator into the differential of the numerator, minus 
the numerator into the differential of the denominator, 
divided by the square of the denominator. 

1. If the denominator is constant, dv = 0, and we 
have, 

vdu du 



dt = 



v 2 w 



SEC. n.J DIFFERENTIALS OF FUNCTIONS. 69 

2. If the numerator is constant, du = 0, and we have, 

. udv 
dt ; 

and under this supposition, t is a decreasing function of 
v (Art. 7) ; hence, its differential coefficient should be 
negative (Art. 19). 

Differentials of Powers. 

29.* To find the differential of any power of a function. 
First, take any function w n , in which n is a positive whole 
number. This function may be considered as composed 
of n factors, each equal to u. Hence (Art. 27), 

d{u n ) d(uuuu . . . .) du du du du 



(uuuu ....)" U U U 



U n KUUUU . . . .) u u u u 



But as there are n equal factors in the numerator of the 
first member, there will be n equal terms in the second ; 

d(u n ) ndu 

hence, — — - = : 

u n u ' 

therefore, d(u n ) = nu n ~ 1 du. 

T 

1. If n is fractional, denote it by — , and make, 

o 

r 

v = u s , whence, v s = u r ; 

and since r and s are entire numbers, we shall have, 

sv*~ 1 dv = ru r - l du; 
from which we find, 

ru r ~ l .. ru r ~ l T 
a v = r du =z du ; 

sv s ~ l : ( ,_!) 

su* 



70 DIFFERENTIAL CALCULUS. [SEC. II 

or, by reducing, 

r 

V - — 1 

dv = - u s du\ 

s 

which is obtained directly from the function, 

d(u n ) — nu n - l dUj 

T 

by changing the exponent w to - • 

/ & 

2. If the fractional exponent is one-half, the function 
becomes a radical of the second degree. We will give 
a specific rule for this class of functions. 

Let v = it?, or, v = -\/u; 

then, dv = -u 2 du = -u * du = 

o o 



2 ""-.a" 2^' 

that is, 

7%e differential of a radical of the second degree, is 
equal to the differential of the quantity under the sign 
divided by twice the radical. 

3. Finally, if n is negative, we shall have, 

1 
u~ n = — , 









u n ' 






from 


which we have 


(Art. 


29), 








<*(«-) = 4~) 


! = - 


- d(u n ) 
u 2n 


= 


nu n - l du 

U 2n 1 


and, 


by reducing, 












£(«-") 


= - 


- nu~ n - 


l du\ 


hence, 



SEC. II. J DIFFERENTIALS OF FUNCTIONS. 71 

The differential of any power of a function, is equal 
to the exponent multiplied by the function raised to a 
2>ower less one, multiplied by the differential of the func- 
tion. 

Formulas for differentiating Algebraic Functions. 

1. d{a) = (Art. 17.) 

2. d(ax) = adx (Art. 18.) 

3. d(x + y) = dx + dy (Art. 26.) 

4. d(x — y) = dx — dy . . . . . . (Art. 26.) 

5. d(xy) = xdy + ydx (Art. 2T.) 

6 ,*(*) = ^J* (^.^ 

7. d(x m ) = mx m ~ l dx (Art. 30.) 

dx 

8. d(<i/x) = Z~r (-Art. 30—2.) 

9. c7(«-«) = x-e~ l dx . . . (Art. 30 — 3.) 

s 

EXAMPLES. 

Find the differentials of the following functions: 

1. u = ax — y. du — adx — dy. 

2. u = a 2 # 2 + z. du = 2a*xdx + cfe. 

3. u = bx 2 — y 3 + a. du — 2bxdx — 3y 2 dy. 

4. u = ax 2 — bx 3 + x. du = (2ax — 3&e 2 -f l)dx. 

5. u =z cy 2 — x 2 + ay 2 . c?w = 2[(c -f a)ydy — xdx.] 



?2 DIFFERENTIAL CALCULUS. [SEC. II 

6. u = xyz, du = yzdx + xzdy + xydz. 

7. u = 2/ 2 — ^ 2 — 8 #z 5 . ^ = 2 (ydy — 20az 4 6fe.) 

8. u = 3« 5 # n . c?w = 3na 5 x n - l dx. 

9. u = — 2ax~ 5 - 5 + 46 2 a 3 . efe = 2^ + 6b 2 x 2 )dx. 

10. m = 5& 5 — 2ay — b 2 . du = 25x*dx — 2ady. 

11. u == jc rt — cc 3 + 46. <#w = (ras 71 - 1 — 3# 2 )cfe. 

12. w = ax(x 2 + 35). c?w = 3a(x 2 + J)efe. 

13. w == (x 2 + a) (x — a). du = (3# 2 — 2a& + a)cfo. 

14. u = x 2 y 2 z 3 . du = 2xy 2 z 3 dx + 2x 2 z 3 ydy + 3x 2 y 2 z 2 dz. 

die = ax(5x 3 -\-2a)dx, 



15. 


w 


= 


a# 2 (sK 3 + a). 


16. 


w 


= 


03 


17. 


m 


= 


a 


5 - 2y 2 


18. 


w 


z= 


1 


19. 


zj 


= 


1 

x~ n = — . 

x n 



, ydx — sedft/ 

* = -^ 



(ft - 2y 2 ) 2 

— dx 
a 2 " 

— ndx 



x n + 
20. Find the differential of u in the equation, 



u = ya 2 — # 2 . 



Put, a 2 — x 2 = y ; then, ^ = yy ; and (Art. 30 — 2), 



du = dy 



SEC. II.] DIFFERENTIALS OF FUNCTIONS. 



73 



But, dy — — 2xdx ; then, substituting for y and dy, 
their values, we have, 



du 



— 2xdx 



— xdx 



2 y/a? — x 2 <y/a 2 — x 2 



21. u =i t/2ox + x 2 
1 



22. u = 



23. u = 



Vi 



du = 



du 



x + -v/T~— x 2 
24. w = (a + t/x) 3 - 



cfe = 



(a + x)dx 
i/2ax + cc 2 

(i - x 2 Y 

dx 



y/l-X 2 (x+<y/l — X 2 ) 2 

du = 3 ( a+ V*) 2dx _ 

2-y/x 



25. u = 



a 2 — x 2 



du = 



a 4 + a 2 x 2 + a 4 
(a 4 +a 2 % 2 +fl 4 ¥(a 2 --sc 2 ) — {a 2 —x 2 )d{a* + a 2 x 2 +x*) 



or, 



cfe = 



(a 4 -fa 2 a 2 +a 4 ) 2 

— 2cc(2a 4 + 2a 2 a; 2 — x *)dx 
(a 4 -f a 2 cc 2 + cc 4 ) 2 



26. w = -/a 2 + x2 X V^M- y 2 



du = 



27. m 



28. u 



x* 



(p2 + y *)xdx + (a 2 + % 2 )y^y 
y/a?~+ x 2 y/b 2 + y 2 



(1 + a)" 

1 + a; 2 
1 — a; 2 ' 



du = t 



(1 + a)* +1 
4#dcc 

(1 -a 2 ) 2 ' 



74 DIFFERENTIAL CALCULUS. [SEC. 1L 

29. u = g + y . du -- < dx + d y) - (* + y)*dz 

Z 3 3 4 



30. u = v^" + ^+ yT-^, 



du = — 



(1 + yl - a;2)^ 



Differential of a particular binomial. 
30.— 1. Let u = (a + bx n ) m . 
Put a + bx n ■- y ; then, u = y m \ and (Art. 30), 

du = my m - l dy. 
But, from the first equation, 

dy = nbx n ~ l dx; 
substituting for y and dy their values, we have, 
du = mnb(a + bx n ) m - l x n ~ l dx; 

that is, to find the differential of a binominal ^unction 
of this form, 

Multiply the exponent of the parenthesis, into the ex- 
ponent of the variable toithin the p>arenthesis, into the co- 
efficient of the variable, into the binomial raised to a 
power less 1, into the variable within the parenthesis 
raised to a power less 1, into the differential of the va- 
riable. 

Rate of change of the Function. 

31. What is the rate of change in the area of a square, 
when the side is denoted by the independent variable? 
We have seen (Art. 2l) that the differential coefficient, 

— > denotes the rate of change in the function u, cor- 
dx 



SEC. II.] DIFFERENTIALS OF FUNCTIONS. 75 

responding to the change dx, in the value of x\ and that 
in all equations, except those of the first degree, this rate 
will be variable, and a function of x (Art. 23). 

Let x denote the side of a square, and u its area ; then, 

u = x J y and — = 2x; 
dx 

hence, the rate of change in the area of a square is equal 
to twice its side ; that is, if the side of a square is denoted 
by 1, the rate of change in the area will be denoted by 2; 
if the edge is denoted by 5, the rate of change will be 10; 
and similarly for other numbers. 

2. What is the rate of change in the volume of a cube, 
when its edge is the independent variable? 

Let x denote the edge of a cube, and u its volume ; 
then, 

u = # s , and — = 3# 2 ; 
dx 

hence, the rate of change in the volume, is three times 
the square of its edge. If the edge is denoted by 1, the 
rate of change in the volume will be denoted by 3 ; if the 
edge is denoted by 2, the rate of change will be 12; if 3, 
the rate will be 27 ; and similarly, when the edge is denoted 
by other numbers. 

Find the rates of change in the following functions: 

3. u ~ Sx A — 3x 2 — 5x + a. A. 32z 3 — 6x — 5. 
What will express the rate for 

x= 1, x = 2, x— 3? 

4. u = (x* + a) (3z 2 + b). A. Ux* + Sx 2 b + 6ax 

Find the rate for, 

x = 1, x = 2. 



76 DIFFERENTIAL CALCULUS. [SEC. II. 

1 1 

1 - a ' T (i _ X )2 

What is the rate for, a; = 0, x z= 4, x = — 1 * 
6. w = (ace + a 2 ) 2 . A 2(aa; + x 2 ) (a + 2#). 

What is the rate for, a? = 0, 05=1, x = 3 ? 



2 



a; 1 

^ «# __ m ji ■ 

" x + -/I - a 2 ' V" 1 - « 2 (* + i/l - « 2 ) 

What is the rate for, x = 0, cc = 1 ? 

Hence, to find the rate of change for a given value of 
the variable : Find the differential coefficient, and substi- 
tute the value of the variable in the second member of the 
equation. 



Partial Differentials. 
32. If we have a function of the form, 

u = ffa'y) (*•). 

the equation denotes that u is a function of the two 
variables, x and y. If we suppose either of these, as y, 
to remain constant, and x to vary, we shall have, 

*-'/■?*»> < a -> 

if we suppose x to remain constant, and y to vary, we 
shall have, 

$ -/!***> <»> 

The differential coefficients which are obtained under 
these suppositions, are called partial differential coefficients 



SEC. n.] DIFFERENTIALS OF FUNCTIONS. 77 

The first is the partial differential coefficient with respect 
to x, and the second with respect to y. 

33. If we multiply both members of Equation ( 2 ) 
by dx, and both members of Equation ( 3 ) by dy, we 
obtain, 

-£ dx = /'(as, y)dx, and -£ dy = f"(x, y)dy. 



The expressions, 



du , du ._ 

-r=- dx, -=- ay, 



are called, partial differentials * the first a partial differ- 
ential with respect to x, and the second a partial differ- 
ential with respect to y ; hence, 

A partial differential coefficient is the differentia^ 
coefficient of a function of two or more variables, wider 
the supposition that only one of them has changed its 
value ; and, 

A partial differential is the differential of a func- 
tion of two or more variables, under the supposition that 
only one of them has changed its value. 

If we suppose both the variables to undergo a change 
at the same time, the corresponding change which take> 
place in u, is called, the total differential. If we extend 
this definition to any number of variables, and assume 
what may be rigorously proved, viz. : 

That the total differential of a function of any number 
of variables is equal to the sum of the partial differ- 
entialsj 



78 DIFFERENTIAL CALCULUS. [SEC. II. 

we have a general formula applicable to every func 
tion of two or more variables. 



EXAMPLES. 



1. Let u = x 2 + y 3 — z ; then, 

du 

— dx = 2xdx, 1st partial differential , 

OCX 

pdy = Sy 2 dy, 2d " K 

€ ^dz = - dz, 3d « « 

hence, rfw == 2ax&2 + Sy 2 dy — dz. 



2. Let u z=z xy ; then, 

— dx — ydx, 
dx 

du _ 
hence, <#w = y^fe + a%?y. 



3. Let w = x m y n ; then, 

-=- dx — mx m ~ * y n dx, 
dx 

-=- dy = /zy n ~ 2 & m dy ; hence, 

c?u = mx m ~ l y n dx + ny n ~ l x m dy = x m -' i y n - l (mydx + rvxdy) 



•*EC. II.] DIFFERENTIALS OF FUNCTIONS, 79 

X 

4. Let u = -: then. 

y 

du ., dx 
— dx — — , 
dx y 

du 7 xdy 

ydx — xdy 
hence, du = -• 

y 2 



5. Let u = - — : == ay(# 2 + y 2 ) 2 ; then, 

y cc 2 + y 2 

c?w T ayxdx 
—- dx = s , 

' >2 1 a/ 2\2 



(a 2 + y 2 ) 



d^^ # ac?y ay 2 dy 

dy 



dy = 3, 

(a 2 + y 2 f (a 2 + y 2 ) 2 



f .. ayxdx — ax 2 dy 
hence, #w = 3— • 

(x 2 + y *y 

6. Let u = #y^ ; then, 

du = yztfefc + £2fc?y + #y^fe + xyzdL 



SECTION III. 

INTEGRATION AND APPLICATIONS. 

34. An Integral is a functional expression, either al- 
gebraic or transcendental, derived from a differential. 

Differentiation and Integration are terms denoting 
operations the exact converse of each other. 

Differentiation is the operation of finding the differ- 
ential function from the primitive function. 

Integration is the operation of finding the primitive 
function from the differential function. 

Rules have been found for the differentiation of every 
form which a function can assume. Hence, in the Differ- 
ential Calculus, no case can occur to which a known rule 
is not applicable. In the Integral Calculus it is quite 
otherwise. 

In returning from a known differential to the integral 
from which it may have been derived, we compare the 
differential expression with other expressions which are 
known to be differentials of given functions, and thus 
arrive at the form of the integral, or primitive function. 
The main operations, therefore, of the Integral Calculus, 
consist in transforming given differential expressions into 
others which are equivalent to them, and which are differ- 
entials of known functions ; and thus deducing formulas 
applicable to all similar forms. 

The integration is indicated by placing the sign / 



SEC. III.] INTEGRATION. 81 

before the expression to be integrated. It is equivalent 
to "integral of"; thus, 

J 2xdx = x 2 , 

is read: "Integral of 2xdx, is equal to £ 2 ." 

Integration of Monomials. 

35. The differential of every expression of the form, 

u = x m , is du = mx m ~ 1 dx (Art. 30), 

which has been found by multiplying the exponent into 
the variable raised to a power less one, into the differ- 
ential of the variable. 

If, then, we have a differential expression, of the form, 

mx m ~ l dx, or, x m dx, 

we can find its integral by reversing the above rule ; that 
is, to find the integral of such an expression, 

Add I to the exponent of the variable, and then divide 
by the new exponent into the differential of the variable* 

EXAMPLES. 

Find the integrals of the following differential expressions • 

1. If du — 2xdx. I du = — = x 2 . 

J 2 x dx 



2. If du = Sx 2 dx, I du = =- 

J 3 y dx 



Sx 3 dx 

- j- = x 3 . 

3 X dx 



* This rule applies to every case of a differential monomial of the 
form, Aa^dx, except that in which m is — 1 (Art. 90). 



82 DIFFERENTIAL CALCULUS. [SEC III 



3. 



If du = aTdx, J du 



x^+^dx x™** 



(m + l)dx m + 1 

ar- 3 + l dx J_ 

~2dx ~ ~ 2x 2 ' 



4. If du — x - 3 dx, I du = — 

5. If du = x 3 ^/xdx, J du = / x 2 dx = -x* y/x. 

36. We have seen, that the differential of the product 
of a constant by a variable, is equal to the constant multi- 
plied by the differential of the variable (Art. 18). Hence, 
the integral of the yyroduct of a constant by a differ- 
ential^ is equal to the constant midtiplied by the integral 
of the differential / that is, 

/ax m dx = a I x m dx = a — — - x m + \ 
J m + 1 

Hence, if the expression to be integrated has one or 
more constant factors, they should, at once, be placed as 
factors, without the sign of the integral. 

37. It has been shown that the differential of the 
sum or difference of any number of variables is equal to 
the sum or difference of their differentials (Art. 26). 
Hence, if we have a differential expression of the form, 

du = 2ax 2 dx — bydy — z 2 dz; we may write, 
J du = 2ajx 2 dx — bjydy —Jz 2 dz; or, 

y» 2 b * 3 

du = -ax 3 y 2 — — ; that is, 
o 2 o 

The integral of the algebraic sum of any number of dif- 
ferentials is equal to the algebraic sum of their integrals. 



SEC. III.J INTEGRATION 83 



Correction — Indefinite— Particular — and Definite Integrals. 

3§. It has been shown that every constant quantity 
connected with a variable by the sign plus or minus, dis- 
appears in the differentiation (Art. 17); that is, 

d(a + x m ) = dx" 1 = mx m - l dx. 

Hence, the same differential may have several integral 
functions differing from each other by a constant term. 
Therefore, in passing from a differential to an integral 
expression, we must annex to the first integral obtained, 
a constant term, to compensate for the constant term 
which may have been lost in the differentiation. 

For example, it has been shown in Art. (22), that, 

~ = a, or, dy = adx, 

is the differential equation of every straight line which 
makes with the axis of abscissas an angle whose tangent 
is a. Integrating this expression, we have, 



/ dy = a J dx (1.) 



or, y = ax; 

or, finally, y = ax + C . . ... (2.) 

If, now, the required line is to pass through the origin 
of co-ordinates, w T e shall have, for 

x = 0, y = 0, and consequently, C = 0. 

But if it be required that the line shall intersect the 



84 DIFFERENTIAL CALCULUS. [SEC. III. 

axis of Y at a distance from the origin equal to + b 
we shall have, for 

x = 0, y = + 5, and consequently, (7 = + J ; 

and the true integral will be, 

y = ax + b ( 3.) 

If, on the contrary, it were required that the right line 
should intersect the axis of ordinates below the origin, 
we should have, for 

x z= 0, y = — J, and consequently, C = — b\ 

and the true integral would be, 

y =z ax -— b (4.) 

The constant (7, which is added to the first integral, 
must have such a value as to render the functional equa 
tion true for every possible value that may be attributed 
to the variable. Hence, after having found the first integral 
equation, and added the constant (7, if we then make tlie - 
variable equal to zero, the value which the function assurneo | 
will be the true value of C. 

1. An indefinite integral is the first integral obtained, 
before the value of the constant C is determined. 

2. A particular integral is the integral after the value 
of C has been found. 

3. A definite integral is the integral corresponding to I 
a given value of the variable. 

Thus, Equation (2) is an indefinite integral, because, so 
long as C is undetermined, it will be the equation of a 



SBC. III.] INTEGRATION. 85 

system of parallel straight lines. Equations ( 3 ) and ( 4 ) 
are particular integrals, because each belongs to a par- 
ticular line. 

Origin of the Integral. 

39. The origin of an integral function is its zero value. 
The value of the variable corresponding to the origin of 
the integral, is found by placing the second member of 
the equation expressing the particular integral, equal to 
zero, and finding therefrom the value of the variable. 
Thus, if in Equation (3), we make y = 0, we have, 

ax + b — 0, and x — — -> 

a 

which shows that the origin of the function y (that is 
y = 0), is on the side of negative abscissas, and at a dis- 
tance from the origin equal to ■ In Equation (4), it is 

at a point whose abscissa is -• 

a 

Integration between limits. 

40. Having found the indefinite integral, and the par- 
ticular integral, the next step is to find the definite in- 
tegral ; and then, the definite integral between given limits 
of the variable. 

Let us take the particular integral found in Equation (3), 

y = ax + b. 

If it is required to find the value of the function ?/, for 
a given value of the variable cc, as, x = x', y will be- 
come a constant for this value, and we shall have, 

y' = ax' + b (5.) 

which is a definite integral. 



86 



DIFFERENTIAL CALCULUS. 



[SEC lit. 




If we wish the value of the function corresponding to a 
second abscissa, x = x f \ we shall have, 

y" = ax" + b (6.) 

If we subtract Equation (5) from Equation (6), we have, 

y" - y' = a(a5" - x') . . . . ( 7.) 

which is the definite integral of y, taken between the lim. 
its, x = #', and x = x" . 

If, x' = OP, and a" = 0$; then, 
y' = PJf, and y" = (^ hence, 
y» - y' = a{x" - x') = JSTE ; 

Therefore : The integral of a func- 
tion, taken between two limits, indi- 
cated by given values of x, is equal to the difference oj 
f he definite integrals corresponding to those limits. 

Let us now explain the language employed to express 
these relations. The modified form of Equation ( 1 ), 

J (dy) x -& == a J dx, 
is read: "Integral of y, when x is equal to «';" an>d 

J (dy) m ~x» = a J dx, 
is read : " Integral of y, when x is equal to x" ; " and 



f(dy) == a J dx, 



is read : Integral of the differential of y, taken between 
the limits, x' and x"\ the least limit, or the limit correspond 
ing to the subtractive integral, being placed below. 



SEC. III.] INTEGRATION. 8? 



EXAMPLE. 



1. What is the integral of da = 9x 2 dx, between the 
limits x — 1, and x = 3, if in the primitive function 
w reduces to 81, when x = 0. 

J du = J 9x 2 dx = 3sc 3 + (7 ; hence, 
A?w = 3cc 3 + ft 

But from the primitive function, t^ = 81, when a; = 0: 
hence, C = 81, and, 

y*rfw = 3a 3 + 81 (1.) 

y*(*0.-i = 3 + 81 = 84 . . (2.) 
/W»-8 = 81 + 81 = 162 . . (3.) 

3 

y(t?w) = 162 — 84 = 78 . . (4.) 
i 

What is the value of the variable corresponding to the 
origin of the integral (Art. 39) ? 

Making the second member of Equation ( 1 ) equal 0, 

3ic 3 + 81 = 0, or, x — — 3. 

Integration of particular binomials. 
41. To integrate a differential of the form (Art. 30), 
du = (a + &r") m af " 'dx (1.) 



88 DIFFERENTIAL CALCULUS. [SEC. Ill, 

The characteristic of this form is, that the exponent of 
the variable without the parenthesis is less by 1 than the 
exponent of the variable within. 

Put, (a + bx n ) = z ; then, (a + bx n ) m = z m ; and, 

dz 
nbx 1l ~ l dx = dz ; whence, x n ~ l dx = — ; hence, 

nb 

/du = / (a + bx n ) m x n ~ l dx = / — — = ^— — ; 
J K J J nb (m + \)nb ' 

and consequently, 

w == —7 — re — f~ -J- O. 

(m + \)nb 

Hence, to find the integral of the above form, 

1. If there is a constant factor, place it without the 
sign of the integral, and omit the power of the variable 
without the parenthesis and the differential: 

2. Augment the exponent of the parenthesis by 1, and 
then divide this quantity, with its exponent so increased, 
by the exponent of the parenthesis, into the exponent of 
the variable within the parenthesis, into the coefficient of 
the variable. 

EXAMPLES. 

1. f(a + Sx^xdx = ( a+ o 8 f )4 + C; and 

m(a + &*).*«& = g(a + bx^ + O. 
mw(a - 4cx*) 2 x*dx = - ^(a - 4cx*) v + C. 



SEC. III.] INTEGRATION. 89 

Integration by Series 

42. The approximate integral of any function of the 

form, 

du —. Xdx, 

may be found, when X is such a function of cc, that it 
can be developed into a series. Having made the 
development of the function X, in the powers of «, 
by the Binomial Formula, we multiply each term by cfe, and 
then integrate the terms separately. When the series is 
converging, we readily find the approximate value of the 
function for any assumed value of the variable. 

EXAMPLE. 

1. Find the approximate integral of, 

fOu = f-Jt= = /V - rffW 
J J -/l - x 2 J 

in which, X = (1 - x 2 )~*. 

__JL 

Developing, (1 — x 2 ) 2 , by the binomial formula,f 

(i _*)-* = i +v» 2 + Y-l^+i-T'l 3 ' 64 '* 0,5 

multiplying by dx, and integrating, we obtain, 
/■, 1 a 3 , 1 3 ^ , 1 3 5 x 1 , . 

y ^ = * + __ + _.__. + _.._ .__. + Ac. 

* Bourdon, Art. 166. University, Art. 32. 
f Bourdon, Art. 135. University, Art. 104. 



^0 DIFFERENTIAL CALCULUS. [SEC. III. 

from which we obtain an .approximate value of u^ cor- 
responding to any value we may give to x. 



APPLICATIONS TO GEOMETRICAL MAGNITUDES. 
Equations of Tangents and Normals. 

43. We have seen, that if x and y denote the 

d\i 

co-ordinates of every point of a curve, ■— will denote 

ctx 

the tangent of the angle which the tangent line makes 

with the axis of abscissas (Art. 13). This value of — 

v ' i X 

was found under the supposition that the second secant 
point became consecutive with the first; hence, 

Any two consecutive points, mast, at the same time, 
be in the chord, the curve, and the tangent. 

Denote the co-ordinates of the point of tangency, in any 
curve, by x" and y". If through this point we draw any 
secant line, its equation will be of the form, 

y - y" = «(« - «")•* 

If the second point of secancy becomes consecutive with 
the first, we shall have (Art. 13), 



dx r 



a — - 



hence, the equation of the tangent line is, 

y-y"= §£(*--«")• • • • (*•) 



* Bk. I. Art. 20. 



SEC. III.] TANGENTS AND NORMALS, 91 

If, m the equation of any curve, we find the value of 

dy" 

~-f- f , and substitute that value in Equation ( 1 ), the equa- 

(XX 

tion will then denote the tangent to that curve. 
1. By differentiating the equation of the circle, 

x 2 + y 2 = E\ or, x" 2 + y" 2 = iJ 2 , 



, dy" x" * 

wehave ' £r>= "P- 

x" 
hence, y - y" = (» - x") ; 

if 

or, by reducing, yy" + xx" = M 2 . 

2. By differentiating the equation of the ellipse, we 

have, 

dy" B*x" 

dx" ~~ A 2 y' r * 

3. By differentiating the equation of the parabola, we 
have, 

ty" P_ t 
dx" - y" % * 

4. By differentiating the equation of the hyperbola, we 

have, 

dy" B 2 x n 

dx" ~~ Ahj ,r 

Substituting these values, in succession, in Equation ( 1 ) , 
and reducing, we shall find the equation of the tangent 
line to each curve. 

* Bk. II. Art. §. \ Bk. III. Art. 14. % Bk. IY. Art. 8. 



92 DIFFERENTIAL CALCULUS. [SEC. Ill, 

44. The equation of the normal is of the form, 

y-y" = a'(x~tf') .... (l.) 

But since the normal is perpendicular to the tangent, at 
the point of contact, 



1 + aa 



' — n* 



V 



1 dx" 

or, a' = = — -— 

a dy" 



hence, the equation of the normal is, 



y-y" = 



W' {x - X,,) 



(2.) 



By differentiating the equation of the circle, the ellipse, 
the parabola, and the hyperbola, finding in each differ- 

dx" 
ential equation the value of =-^, substituting that 

value in Equation (2), and reducing, we shall find the 
equation of the normal line to each curve. 



Value of tangent, sub-tangent, normal, and sub-normal. 

45. Let P be any point of a 
curve; TP the tangent, TP the 
sub-tangent, PN the normal, and 
BJV the sub-normal. 

Then, in the right-angled tri- 
angle TPB, 

PR= TRx tan PTP = 7!Ex?; 

dx 

hence, TJR = —=- = y— = Sub-tangent. 

O if 

dx 




* Bk. I. Art. 23. 



SEC. III.] TANGENTS AND NORMALS. 93 

46. The tangent TP is equal to the square root of 
the sum of the squares of TR and PR; hence, 



TP= ys Jl+^ = Tangent. 

47. Since TPN is a right angle, RPN is the com 

plement of TPR ; it is therefore equal to PTR, and con- 
sequently its tangent is ■— ; hence, 

CtX 

RN = y-~ = Sub-normal. 
* dx 

48. The normal PN is equal to the square root of 
the sum of the squares of PR and RN~\ hence, 



PN = y yjl + ^ = Normal. 

49. Apply these formulas to lines of the second order, 
of which the general equation is, 

y 2 -: mx -f- nx 2 .* 

Differentiating, we have, 

dy m + 2nx m + 2nx 



d x 2y 2yWc + nx 2 

substituting this value, we find, 

_,_> dx 2(mx + nx 2 ) „ , 

TR = yTy = "k+W = S"b-tangent. 



* Bk. V. Art. 42. 



94 



DIFFERENTIAL CALCULUS. 



SEC. III. 



BN - y-f = — - = Sub-normal. 



p^ 



1=7 y V * + 5& = V maj + wa!2+ i^ m + 2 ^ )2, 



"By attributing proper values to m and n, the abo^e 
formulas will become applicable to each of the conic 
sections. In the case of the parabola, n = 0, and we 
have, 



ZIB z= 2as, 



ZP = ^/m^+ 4a 2 , 



*»■-.?• 



PiV = \/mx + -m 2 . 



Asymptotes. 

50. An asymptote of a curve is a line which continually 
approaches the curve, and becomes tangent to it at an 
infinite distance from the origin of co-ordinates. 

Let AX and AY be the 

co-ordinate axes, and 

y-y"= §£(«-«"), 

the equation of any tangent 
line, as ZIP. 

If, in the equation of the tangent, we make, in succes- 
sion, y = 0, & = 0, we shall find, 




x = AT = x" - y"~ 



,dx" 

w 



— AT) — ti" — cr" y . 

_ AJJ - y x M> 



SEC. III.] ASYMPTOTES. 95 

If the curve CPB has an asymptote RE, it is plain 
that the tangent PT will approach the asymptote RE, 
when the point of contact P, is moved along the curve 
from the origin of co-ordinates, and T and D will also 
approach the; points R and Y, and will coincide with 
them when the co-ordinates of the point of tangency are 
infinite. 

In order, therefore, to determine if a curve have asymp- 
totes, w^e substitute in the values of AT and AD, the 
co-ordinates of the point which is at an infinite distance 
from the origin of co-ordinates. If either of the dis- 
tances AT, AD, becomes finite, the curre will have an 
asymptote. 

If both the values are finite, the asymptote will be 
inclined to both the co-ordinate axes ; if one of the dis- 
tances becomes finite and the other infinite, the asymptote 
will be parallel to one of the co-ordinate axes; and if 
they both become 0, the asymptote will pass through the 
origin of co-ordinates. In the last case, we shall know 
but one point of the asymptote, but its direction may be 

determined by finding the value of ~ , under the sup. 

position that the co-ordinates are infinite. 

51. Let us now examine the equation, 
y 1 = mx + nx 2 , 

of lines of the second order, and see if these lines have 
asymptotes. We find, 

2y 2 — mx 



AT = x- 



m + 2nx m + 2nx* 



96 



DIFFERENTIAL CALCULUS 

?nx + 2nx 2 



[sec. Ill 



AD = y 



mx 



2 2/ 2 ^/mx 4- nx 2 ' 

which may be put under the forms, 



AT = 



— m 



m 
x 



AD = 



m 



+ 2n 



m 

— + n 
x 



and making x = oo, we have, 



77^ 

-4-R = — — , and ^4^ = 
2n 



m 



2-y/rc' 




If now we make n = 0, 
the curve becomes a parabola, 
and both the limits, -4_Z2, AE, 
become infinite ; hence, the 
parabola has no rectilinear 
asymptote. 

If we make n negative, the 
curve becomes an ellipse, and 

AE becomes imaginary ; hence, the ellipse has no asymp- 
tote. 

But if we make n positive, the equation becomes that 
of the hyperbola, and both the values, AH, AE, become 

2jB 2 
finite. If we substitute for m its value, —j- , and fof 

B 2 

?i its value -t-, we shall have, 

A 2 



AH = 



A, 



and 



AE = =b B. 



Hence, of the lines of the second order, the hyperbola 
alone has asymptotes. 



SEC. III.] RECTIFICATION OF CURVES. M 

Differential of an arc. 

52. We have seen that, when the points which limit 
any arc of a curve become consecutive, the chord, the 
arc, and tangent become equal (Art. 43) ; therefore, the 
differential of an arc is the hypothenuse of a right-angled 
triangle of which the base is dx, and the perpendicular 
dy. Hence, if we denote any arc, referred to rectangular 
co-ordinates, by z, we have, 

dz = y/dx 2 + dy 2 . . (1.) or, z = J -y/dx 2 + dy 2 ..( 2.) 

Rectification of a plane curve. 

53. The rectification of a curve is the operation of 
finding its length; and when its length can be exactly 
expressed in terms of a linear unit, the curve is said to 
be rectifiable. . To rectify a curve, given by its equation: 

Differentiate the equation of the curve and find the 

value of dy 2 in terms of x and dx; or of dx 2 in 
terms of y and dy, and substitute the value so found 

in the differential Equation (2). The second member 

will then contain but one variable and its differential; 

the integral will express the length of the arc in terms 
of that variable. 

EXAMPLES. 

1. Find the length of the arc of a circle in terms of 

the radius. The equation of a circle whose radius is 1, 

referred to rectangular axes, when the origin is at the 

centre, is, 

x 2 + y 2 = 1. 



98 DIFFERENTIAL CALCULUS 

Denoting the arc by 2, we have, 

dz = < x / r dx 1 ~+ dy 2 , or, z = J \/dx 2 + dy 
From the equation of the circle, we have, 
xdx + ydy = ; hence, 



[sec hi 






= m 



cfe 2 + 



x 2 dx 2 



/dx /> _ x 

7f^=./(i-* 2 ) ^ 



Developing the binomial factor into a series, by the 
binomial formula,* multiplying by dx, and integrating, we 
have (Art. 42), 

.=/<i-rf) A-^ + p + ^ + aS^+^ + a 

If we suppose the origin of the 
integral to be at i?, the correspond- 
ing value of x w r ill be zero, and 
C -— 0. If now we integrate between 
the limits x = 0, and & = 1, we 
shall obtain the value of the cor- 
responding arc in terms of the radius 1. 

But as, or PiJf, is the sine of the arc EP, denoted 
by z\ and when x — \, z —. 30°; hence, 
i 




30° =/(! - ■*)-** = j + -L, + j^b + &c, 



2.3.2 3 



2.4.5.2 5 



* Bourdon, Art. 135. University, Art. 104. 



* 



SEC. III.] RECTIFICATION OF CURVES. OS 

hence, 

/l 1.1.1 1.3.1.1 1.3.5.1.1 - \ 

30 x6 = 6 ( 2 + m* + tjjrj* + tzejj, + &«•)> 

and by taking the first ten terms of the series, we find. 
*r = 3.1415926. . . , 

a result true to the last decimal figure. 

We have thus found the semi-circumference of a circle 
whose radius is 1, or the circumference of a circle whose 
diameter is 1. 

2. Find the length of the arc of a parabola, whose 
equation is, 





y 2 = 


2px. 






Differentiating and 


dividing 
ydy = 


: by 2, 
pdx, 


we 


have, 


Ld consequently, 


dx 2 = 


$W; 







substituting this value in the differential of the arc, we 
have, 



dz = y/*/ 2 + ^ 2 dy 



= — d vVp 2 + y 2 \ 

developing the radical quantity by the binomial formula, 
and integrating the terms separately, we have, 

" V + 2 " 3 p 2 ~ 2 " 2 " 2 ' 5p 4 + 2 ' 2 ' 2 ' 2 ' 3 ' lp 6 ~~ °7 

LofC. 



100 



DIFFERENTIAL CALCULUS. 



[sec. HL 



If we estimate the arc from the principal vertex, z and y 
will be zero together, and C will be zero. If we make 
y — p, z will denote the length of the arc from the vertex 
to the extremity of the ordinate passing through the focus. 



QUADRATURES. 

54. Quadrature is the operation of finding the area 
or measure of a surface. When this measure can be found 
in exact terms of the unit of measure, the surface is said 
to be quadrable. 

Quadrature of plane figures. 

55. A plane figure is a portion of a plane, bounded by 
lines, either straight or curved. 

Let be the origin of a sys- 
tem of rectangular co-ordinates, 
and oacdeb any line whose equa- 
tion is of the form, 




y =A®) 



(i.) 



O A G D 



If the ordinate Oo, denoted by y, move parallel to itself, 
along OD as a directrix, and so change its value as always 
to satisfy Equation ( 1 ), it will generate the plane surface 
OoacdD, and its upper extremity will generate the line 
oacd. The element, or differential of this surface will be 
any one of the trapezoids, as CcdD, when the ordinates 
Cc and Dd are consecutive. If we denote the surface on 
the left of the ordinate Cc, by s, ds will denote the area 
of the trapezoid. This trapezoid is composed of the rect- 
angle Cd\ and the triangle cd f d\ that is, 

dydx 
ds = ydx + -^— • 



BEC.ni.] QUADRATURES. 101 

But since the product ydx is an infinitely small quantity 

of the first order, and dydx an infinitely small quantity of 

the second order, the latter may be omitted without error 

(Art. 20) ; hence, 

ds = ydx; that is, 

The differential of a plane surface is equal to the or 
dinate into the differential of the abscissa. 

To apply the principle enunciated in the last equation, 
in finding the measure of any particular plane surface : 

Find the value of y in terms of x, from the equation 
of the bounding line; substitute this value in the differ- 
ential equation, and then integrate between the required 
limits of x. 

Nature of the Integral. 

56. To comprehend the true nature of an integral, we 
must examine the differential from which it was derived. 
The differential of a plane surface is, 

ds = ydx. 

If we integrate between the limits x — 0, and x — 01) — a 9 
we write, 



I ds — I ydx — OoacdD ; 



that is, the first member of the equation denotes the sum 
of all the infinitely small rectangles between the limits 
jc — 0, and x = a; the second member, 



/» 



ydx, 
is the same thing under another form ; viz. : it shows that 



102 DIFFERENTIAL CALCULUS. [SEC. Ill, 

every value of y, between the limits y — Oo, and y — B\ 
is multiplied, in succession, into each base denoted by 
dx\ the sum of these products, each of which is ydx, is 
obviously the required area. 

1. Perhaps the relation between the differential and the 
integral, may be more obvious, by observing the figure, 
in which a portion of it is divided into three parts, having 
equal bases. If we bisect each base and draw parallel ordi- 
nates, we shall have six parts ; if we bisect again and draw 
parallel ordinates, we shall have twelve parts; if again, 
twenty-four; and so on. 

Now, there is no difficulty in seeing that each bisection 
doubles the number of parts, and dimmish es the value of 
each part ; and that the sum of the parts will be constantly 
equal to the given area. When, therefore, each part be- 
comes infinitely small, any finite number of them is 0; 
but an infinite number is equal to a finite quantity, viz. : 
to the given area. 

Area of a rectangle. 



57. Let be the origin of a J 

system of rectangular co-ordinates. -° 



A X 



On the axis of Y, take any dis- 
tance OB equal to h. Suppose 
the line h to move parallel to 
itself, along the axis of JT, as a directrix, until it reaches 
the position AG. During its motion, it will generate the 
rectangle OG\ the foot of the line will pass over every 
point in the line OA, and the line itself will occupy every 
part of the rectangle OG. 



SEC. HI.] QUADRATURES. 103 

Since the equation of the line JBG is, 

y = h 

we shall have, for the differential of the surface, 

ds = hdx. 

Integrating between the limits x — 0, and x = 6, and 
observing that C = 0, when x = 0, we have, 

I ds = J hdx = hx = hb; that is, 



7%0 area o/* a rectangle is equal to the product of its 
base by its altitude. 

Area of a triangle. 

58. Let ABC be a right-angled 
triangle, and C the origin of co- 
ordinates. Denote the base AB by 
J, and the altitude CB by h. De- 
note any line parallel to the base by 
y, and the corresponding altitude 
by x. 

If we suppose the base AB to be moved towards the 
vertex of the triangle, along CB as a directrix, and so tc 
change its value, that, 




b : h 



V : v, 



or, 



bx 



it is plain that it will generate the surface of the triangle, 
If we denote the surface by s, we have, 

ds = ydx\ 



104 DIFFERENTIAL CALCULUS. [SEC. III. 

substituting for y its value, and integrating between the 
limits x =: 0, and x = h, we have, 

h 

r 7 b r y b x 2 bh 
Jds = -J xa x = - h - = -; 



that is, The area of a triangle is equal to half the 
product of the base by the altitude. 



Area of the parabola. 

59. Find the area of any portion of the common para- 
bola whose equation is, 

y 2 = 2px\ whence, y = y^2px. 

This value of y being substituted in the differential equa- 
tion (Art, 55), gives (Art. 36), 

fch = fi/2jmkc = V^fx^dx = ^5b* + C; 

2^2px XX 2 , „ 

or, s = — ^~ = ^y + CI 

If we estimate the area from the principal vertex, where 
x = 0, and y = 0, we have, (7=0, and denoting 
the particular integral by s\ we shall have, 

2 

s' = -xy\ that is, 
o 

2%6 ar^a q/ 1 any portion of the parabola, estimated 

2 
from the vertex, is equal to - of the rectangle of the 

o 

abscissa and ordinate of the extreme point. The curve 
?£, therefore, quadrable. 



SEC. III.] QUADRATURES. 105 

1. To find the area of a parabola from the vertex to 
the double ordinate through the focus. We have, for these 
limits, x = and x = \p. Denoting the integral by s", 

we have, / ds = s" = ^p 2 , 



which denotes the area bounded by the curve, the axis, 
and the ordinate; hence, if we double it, we shall have the 
required area; or, 

2*" = ir s ='V=i(22>) 2 ; 

That is, The area is equal to one-sixth of the square 
described on the parameter of the axis. 

2. If the area be estimated from the ordinate through 
the focus, where x = \p, and y = jt>, O must have such a 
value as to reduce the first member to 0: for, this is the 
origin of the integral. 

We have, J ds — \xy + (7; 

and for the particular case of the focus, 

fds = %xipXp + C= ^p 2 + Q\ hence, 

\p* + (7=0; or, C= -ip 2 . 
Hence, the integral from x = \p to any value of x is, 

/2 1 

to 

Area of the circle. 

60. The equation of the circle referred to its centre 
and rectangular axes is, 



y 2 = r 2 — x 2 ; or, y = ^r 2 "— x 2 ; 



106 DIFFEKENTIAL CALCULUS. [SEC. Ill 

hence, the differential equation of the area (Art. 57) is. 
ds = (v^ - — x*)dx .... (1.) 

in which the origin of the area is at the secondary dia* 
meter, where x = 0. 

From Formula JB 3 page 189, we have, 

/ ( VV 2 — x 2 ^dx = -#(r 2 — # 2 ) 2 -h -r 2 / (r 2 — cc 2 )" ^tfe- 

But, by Formula (13), Art. 99, we have, 

f(f _ a.)"*^ = f—M= = sin" 1 - + <7; 
^ ^ V^- a 2 r 

whence, by substitution, we have, 



s = -#(r 2 — sc 2 ) 2 + -r 2 sin — \- C . (2.) 

Estimating the area from the secondary diameter, where 
x = 0, we have, G = 0. 

If we integrate between the limits of x = 0, and 
as = r, we shall have one quarter of the area of the 
circle. When we make x = r, in Equation ( 2 ), the 
first term in the second member becomes ; and in the 

x 
second term, - becomes 1, and the arc whose sine is 1, 

r 

<if 
is 90% which is denoted by -—, to the radius 1 ; hence, 

2i 

T 

/l 1 *If 

ds = -r* sin- 1 1 = -r= x -; or, 

Area of the circle = 41 -r 2 x -1 = rV. 



SEC. m.] QUADRATURES. 10? 

Area of the ellipse. 

61. The equation of the ellipse, referred to its centra 
and axes is, 

A 2 y 2 + B 2 x 2 = A 2 B 2 ; hence, 

y = -jv^rr^i, 

and the differential equation of the area is, 

B i 

ds = ~i(A 2 — x 2 ) 2 dx. 

The second member of this equation differs from the 
second member of Equation ( 1 ), of the last Article, only 

in the constant coefficient -r , and the constant A 2 for 

A 

r 2 , within the parenthesis ; hence, the integral of that 
expression becomes the integral of this, by multiplying it 

by -j , and changing r into A ; that is, 

f ds = a\1 A2 x I) = ~r^ ; hence ' 

Area of ellipse = — - — = A.B.*\ that is, 

The area of an ellipse is equal to the product of its 
semi-axes multiplied by *. 

1. Let Q denote the area of a circle described on 
the transverse axis, and Q' the area of a circle described 
on the conjugate axis; then, 

A?« = Q, and IP* = Q' ; hence, 



108 DIFFERENTIAL CALCULUS. 

A 2 B 2 * 2 = QQ', and 



[sec. in. 



AB« = \/Q~x Q!\ that is. 




A G D E B X 



The area of an ellipse is a mean proportional between 
the two circles described on its axes. 

QUADRATURE OF SURFACES OF REVOLUTION. 

62. Let oacdeb be a plane 
curve, OB the axis of abscis- 
sas, and Oo, Aa, Cc, &c., con- 
secutive ordinates,; then, oa, ac, 
cd, &a, will be elementary arcs. 
The surface described by either 
of these arcs, while the curve 
revolves around the axis OB, will be an element of the 
surface. We have seen, that when the ordinates are con- 
secutive, the chord, the arc, and the tangent, are equal 
(Art. 43) ; hence, the surface described by any arc, as 
ac, is equal to that described by the chord; that is, 
equal to the surface of the frustum of a cone, the radii 
of whose bases are Aa = y, Cc = y + dy, and of which 
the slant height ac = ^/dx 2 + dy 2 . Hence, if we denote 
the surface by s, we have,* 

ds = «(2y + 2y + 2dy) x \y/d& + dy 2 ; 

or, omitting 2dy (Art. 20), 

ds = 2tfy^/dx 2 + dy 2 ; that is, 

The differential of a surface of revolution is equal to 
the circumference of a circle perpendimlar to the axis, into 
the differential of the arc of the meridian curve. 



Leg., Bk. VIII. P. 4. 



SEC. III.] SURFACES OF REVOLUTION. 



109 







Therefore, to find the measure of any surface of revo- 
lution : 

Find the values of y and dy, from the equation 
of the meridian curve, in terms of x and dx; then 
substitute these values in the differential equation, and 
integrate between the proper limits of x. 



Surface of a cylinder. 

63. If the rectangle AC be 
revolved around the side AB, 
DC will generate the surface of 
a cylinder. 

Since the generatrix is parallel 
to the axis AB, its equation will 
be, 

y = J, and hence, dy = 0. 

Substituting these values in the differential equation of 
the surface, we have, 

J ds = J2*yi/'dx 2 + dy 2 = J 2«bdx = 2«bx + C. 

If we suppose A to be the origin of co-ordinates, 

(7=0, and integrating between the limits x — and 
x = A, we have, 

s = 2b<th\ 

that is, The measure of the surface of a cylinder is equal 
to the circumference of its base into the altitude. 




110 DIFFERENTIAL CALCULUS. [SEC. Ill 

Surface of the cone. 

64. If the right-angled triangle 
CBA be revolved around the axis 
AC, CB will generate the convex 
surface of a cone. 

If we suppose C to be the origin 
of co-ordinates, the equation of BO 
will be, 

y = ax, and dy — adx. 

Substituting these values in the differential equation of 
the surface, we have, 

fds — J2«axy/dx 2 + a 2 dx 2 = J2«axdx^T+ a 2 + (7, 

(Art. 35) = <fax 2 i/T + « 2 + O. 

Estimating the surface from the vertex, where x = 0, 
we have, (7=0, and 

s = *ax 2 i/l + a 2 . 

If we make x = h = AC, and BA = J, we have, 
a = 7 , and consequently, 



= 'l »s/> 



, b 2 2*b+0+ b 2 n , BC 

1 + -To = ^ = 2* b X — - • 

A 2 2 2 



that is, The convex surface of a cone is equal to the 
circumference of the base into half the slant height 



i 



SEC. III.] SURFACE OF THE SPHERE. Ill 

Surface of the sphere. 

65. To find the surface of a sphere. The equation 
of the meridian curve, referred to the centre, is, 

x 2 + y 2 = i?. 

By differentiating, we have, 

xdx + ydy = ; 
hence, 

xdx .. 7 , x 2 dx 2 

*y = - — ' and dy = ~^~' 

Substituting for dy 2 its value, in the differential of the 
surface, which is, 

ds = 2«y<\/dx 2 + dy\ 
we have, 



fds = flvysjdx 2 + — 2 dx 2 = J^Rdx = 2* Ex + 0. 

if 

If we estimate the surface from the plane passing through 
the centre, and perpendicular to the axis of -3T, we shall 
have, 

5 = 0, for x — 0, and consequently, (7=0. 

To find the entire surface of the sphere, we must inte- 
grate between the limits x — + i?, and x = — H, 
and then take the sum of the integrals, without reference 
to their algebraic signs ; for, these signs only indicate the 
position of the parts of the surface with respect to the 
plane passing through the centre. 

Integrating between the limits, 

x = 0, and x = +- H 9 



112 DIFFERENTIAL CALCULUS. [SEC. ITL 

we find, s = 2tfH 2 ; 

and integrating between the limits x = 0, and x = — H, 
there results, 

s = — 2*i? 2 ; 
hence, 

Surface = 4<ri? 2 = 2«R x 2i? ; 

that is, Equal to four great circles, or equal to the 
curved surface of the circumscrioi?ig cylinder. 

1. The two equal integrals, 

s — IxR 1 , and e = — 2^rJ? 2 , 

indicate that the surface is divided into two equal parts 
by the plane passing through the centre. 

Surface of the paraboloid. 

66. To find the surface of the paraboloid of revolution. 
Take the equation of the meridian curve, 



y 2 = 2px, 




which being differentiated, gives, 




dx = V—z- , and dx 2 = 
P 


y 2 fy 2 
p* 



Substituting this value of dx in the differential of the 
surface, (Art. 62), we have, 



ds = 2cry^/(^±^)rfy = jyd VA /j/*~+ p. 



SEC. III.] SURFACE OF THE ELLIPSOID. 115 

But we have found (Art. 41), 



JjydyV¥+p 2 = ^(y 2 + P 2 )> + C; 



hence, 



2nt 1 

= ^(y 2 + p 2 ) 2 + G. 



If we estimate the surface from the vertex, at which 
point y = 0, we shall have, 

= ^+0, whence, C = - 2 -f; 

and integrating between the limits, 

y = 0, and y = ft, 



we have, 



• = ^'+P s r-A 



Surface of the ellipsoid. 

67, To find the surface of an ellipsoid described by 
revolving an ellipse about the transverse axis. 

The equation of the meridian curve is, 

Ahf + B 2 x> = A 2 B?, 
whence, 

, B 2 xdx B xdx 

d V = - -72 ■ 



A 2 V A y/A*-x*' 



114 



DIFFERENTIAL CALCULUS. 



[SEC. III. 



substituting the square of dy in the differential of the 
surface, and for y its value, 



j^p 



ar 5 . 



we have, 



B 



ds = 2if-^dx<^J*^{A*- B^x*; (1.) 



hence, fds = 2*— ^/X 2 - £ 2 fdx\/-jrZT 



B< 



— a;' 



B 



Put, 2'7r-j^y'^l 2 — i? 2 = J9, a constant quantity; 



and 



4 2 - ^ 2 



jR 2 , also a constant, 



and we have, 



fds = Dfdx^fi 2 - x\ 




i<: A 



B D 



With C, the centre of the 
meridian curve, and the radius 
i?, describe a semi-circle. Then, 

/ dx t/1& — cc 2 , is a circular 

segment of which the abscissa 
is #, and radius JR. 

If, then, we estimate the surface of the ellipsoid fron. 
the plane passing through the centre, and estimate the 
area of the circular segment from the same plane, any 
portion of the surface of the ellipsoid will be equal to 
the corresponding portion of the circle, multiplied by the 
constant JD. Hence, if we integrate the expression, 



SEC. III.] CUB A TU RE OF VOLUMES. 115 

fdx^/B 2 - X 2 , 

between the limits x = 0, and x = A, we shall have 
the area of the segment CGFB, which denote by D'. 
Hence, 

| surface ellipsoid — D x D f ; and 

Surface = 2D x -Z>'. 

1. If we make ^4 = B, in Equation (1), the ellipsoid 
becomes a sphere, and we have, 

s = / 2<fBdx = 2rfjRa5 + ft 

If we estimate the surface from the plane passing through 
the centre, (7=0, and integrate between the limits 
x = 0, and s» = i£, we have, 

£ surface of sphere = 2tfi? 2 ; hence, 
Surface = ±«B 2 . 

CUBATUKE OF VOLUMES OF EEVOLUTIOIST. 

6§. Cubature is the operation of finding the measure 
of a volume. When this measure can be found in exact 
terms of the measuring cube, the volume is said to be 
cubable. 

69. A volume of revolution is a volume generated by 
the revolution of a plane figure about a fixed line, called 
the axis. 

If the plane figure OoacdebB, be revolved about the 
axis of JT, it will generate a volume of revolution. 



116 



DIFFERENTIAL CALCULUS. 



[sec. in. 




A G D E B X 



Let us suppose the ordinates Aa, Cc, Dd, &c, to be 
consecutive. During the revo- 
lution, any element of the sur- 
face, as, AacC, will generate 
the frustum of a cone, of which 
the radii of the bases are 
Aa = 2/, Go — y + dy, and 
the altitude, AG — dx. This 
frustum will be an element of the volume, and will have 
for its measure,* 

lb/ 2 + (y + dy) 2 + y(y + dy)]dx. 

If we denote the volume by T 7 ", develop the terms 
within the parenthesis, multiply by dx, and then reject 
all the terms containing the infinitely small quantities of 
the second order (Art. 20), we shall have, 

dV = <xy 2 dx. 

The area of a circle described by any ordinate y, is 
*y 2 \ ; hence, The differential of a volume of revolution is 
equal to the area of a circle perpendicular to the axis into 
the differential of the axis. 

The differential of a volume generated by the revolution 
of a plane figure about the axis of Y", is irx 2 dy. 

70. To find the value of V for any given volume : 

Find the value of y 2 in terms of x, from the equation 
of the meridian curve / substitute this value in the differ- 
ential equation, and then integrate between the required 
limits of x. 



* Leg., Bk. VIII. P. 6. 



f Leg., Bk. V. Prop. 16. 



SEC. III. j CUBATURE OF VOLUMES. 



11? 



EXAMPLES. 

1. Find the volume of a right cylinder with a circular 
base, whose altitude is h and the radius of whose base is r. 

We have for the differential of the volume, 
dV = fy 2 dx; 
and since y z=z r, we have, 



fdV = f«r 2 dx; 



integrating between the limits x — 0, and x = h, 

h 

f dV — V ' = *r 2 x = <xr 2 h ; that is, 



The measure of the volume of a cylinder is equal to 
the area of its base multiplied by the altitude* 

2. Find the volume of a right cone with a circular base, 
whose altitude is A, and the radius of the base, r. 

If we suppose the vertex of the cone to be at the origin 
of co-ordinates, and the axis to coincide with the axis of 
abscissas, we shall have, 

y = ax, or, y = |a, and y 2 = j^a?; 
substituting this value of y 2 , we have, 

* Legendre, Bk. VIII. Prop. 2. 



118 DIFFERENTIAL C A J. CULLS. [SEC. III. 

Integrating between the values x = 0, and x = h 9 

h 

JdV = V== ^3" = **■ X g> that is, 



7%£ measure of the volume of a cone is equal to the 
area of the base into one-third of the altitude* 

3. To find the volume of a prolate spheroid, f 
The equation of the meridian curve is, 

B 2 
A*y 2 + B 2 x 2 = A 2 B 2 ; hence, y 2 = -p(^ 2 - « 2 )« 



and 



JB 2 

dV = m—^A? — x 2 )dx; hence, 






If we estimate the volume from the plane passing 
through the centre, we have, for x = 0, T^ = 0, and 
consequently, (7=0; and taking the integral between 
the limits x = 0, and x = A, we have, 



y^F = jW x -4 ; 



which is half the volume ; consequently, the entire volume, 



2V = %eB* X 2J. 



Legendre, Bk. VIII. Prop. 5. f Bk. VI. Art. 37. 



SBC. III.] CUBATUKE OF VOLUMES. 119 

But, r.B 2 expresses the area of a cirele described on the 
conjugate axis, and 2 A is the transverse axis; hence, 

The oolume of a isolate spheroid is equal to two-third*, 
vf the circumscribing cylinder. 

1. If an ellipse be revolved around the conjugate axis, 
it will describe an oblate spheroid, and we shall have, 



I dV '= / <xx 2 dy; 



substituting for # 2 , and integrating, we have, 
2V= \«A* X 2jB; 

that is, two-thirds of the circumscribing cylinder. 

2. If we compare the two results together, we find, 

oblate spheroid : prolate spheroid : : A : B. 

3. If we make JB = A, the ellipsoid becomes a sphere 
whose diameter is the transverse axis. Then, 

2V= JtB* X D = lrJ5»j 

that is, Equal to two-thirds of the circumscribing cylinder, 
or to one-sixth of <x into the cube of the diameter. 

4. Find the volume of a paraboloid. The equation of 
the meridian curve is, 

y 2 = 2px; hence, 
dV — 2tpxdX) and V= *p&. 

If we estimate the volume from the vertex, (7=0. If wc 
integrate between the limits x = 0, and x = h, and de- 
signate by #, the ordinate corresponding to the abscissa 
x = A, we have, 



120 



DIFFERENTIAL CALCULUS. 



[sec. hi 




V~ *ph 2 — *b 2 x -; that is, 
equal to half the cylinder having the same base and altitude 

Prism and Pyramid. 

1. Let ABCDE be any polygon, and FH a line per- 

pendicular to the plane of the base. If v 

the polygon move along the line FH, 

parallel to itself, it will generate a prism. 

If we denote the volume by T 7 , the area of ( 

the base by 5, and the indefinite line HF 

by x> we shall have, 

dV= bdx. 

and, integrating between the limits x =■ 0, and x == A, 

h 

I dV = I bdx = bxx = bxh. 



2. If we suppose the base so to vary, as it moves along 
the line FH, as to bear a constant ratio to the square of 
its distance from the point F, it will generate the volume 
of a pyramid, of which F is the vertex and ABODE the 
base.* If we denote the variable generatrix, at any point, 
by 2/, and its distance from the vertex by #, we have, 

dV — ydx. 

But, b : y : : h 2 : x 2 \ hence, y == — x x 2 ; 

lb 

therefore, dV= ^ X x 2 dx: 

hr 

and integrating between the limits x == 0, and x = A, we have, 
h 

f dV — y- x 2 dx = T^X — = J X -• 
^ /* 2 «/ h 2 3 3 



* Legendre, Bk, VII. P. 3. Cor 1. 



SECTION IV. 

SUCCESSIVE DIFFERENTIALS SIGNS OF DIFFERENTIAL CO 

EFFICIENTS FORMULAS OF DEVELOPMENT. 

Successive Differentials. 

71. If u denotes any function, and x the independent 
variable, we have seen that the differential coefficient P, 
is, in general, a function of x (Art. 23). It may there- 
fore be differentiated, and a new differential coefficient 
will thus be obtained, which is called the second differ- 
ential coefficient. 

In passing from the function u to the first differ- 
ential coefficient, the exponent of x is diminished by 1, in 
every term where x enters (Art. 30); hence, the relation 
between the primitive function u and the variable as, is 
different from that which exists between the first differen- 
tial coefficient and x. Hence, the same change in x. 
will occasion different degrees of change in the primitive 
function and in the first differential coefficient. 

The second differential coefficient will, in general, be a 
function of cc, exhibiting a still different relation ; hence, 
a new differential coefficient may be formed from it, 
which may also be a function of x\ and so on, for suc- 
ceeding differential coefficients. 



122 DIFFERENTIAL CALCULUS. [SEC. IV. 

If we designate the successive differential coefficients by 
p, q, r, s, &c, 



we shall have, 










du 


dp 
dx 


% 


fi = r, Ac; 


and 


du = pdx, 


dp — 


qdx, 


dq = rdx. 





But the differential of p may be obtained by differ- 
entiating its value — , regarding the denominator dx as 
constant; we therefore have, 

Jdu\ 7 d 2 u y 

AtJ = * or ' w = dp > 

substituting for dp its value, and dividing by dx, 

d 2 u 

The notation, d 2 u, indicates that the function u has been 
differentiated twice ; it is read, second differential of u. 
The denominator dx 2 , denotes the square of the differ- 
ential of x, and not the differential of x 2 . It is read; 
differential of a;, squared. 

If we differentiate the value of q, we have, 

Jd 2 u\ _ d 3 u - 

d \d*l = *' ° r ' 3* = * ; 

d^u 
hence. -7-= = r, &c. ; 



SEC. 1V.J SUCCESSIVE DIFFERENTIALS. 123 

and in the same manner we may find, 

cPu 
The third differential coefficient, -=— , is read : third 

dx 3 

differential of it, divided by dx cubed ; and the differ- 
ential coefficients which succeed it are read in a similar 
manner. 

Hence, the successive differential coefficients are, 
du dhc dhc dhc „ 

* = » a* - ft a? = r ' ^ = s ' &c " 

from which we see, that each differential coefficient is 

derived from the one that immediately precedes it, in the 

same way as the first is derived from the primitive func- 
tion. 

The differentials of the different orders are obtained by 
multiplying the differential coefficients by the correspond 
ing powers of dx ; thus, 

-zr- dx = 1st differential of u. 
dx 

d 2 u 

—-dx 2 = 2d differential of u. 

dx 2 



■ d n u 

-=— dx n = nth. differential of tu 
dx n 



124 DIFFERENTIAL CALCULUS. [SEC. IV 

EXAMPLES. 

1. Find the differential coefficients in the function, 
u == ax 3 . 

— = 3ax 2 ss », 



<fc 2 



= 6acc = y, 



d 3 w . 



2. Find the differential coefficients in the function, 

u = oas*. 
The first differential coefficient is, 



cfo 



== nax n ~ l . 



Since ??, a, and dx, are constants, we have for the 
second differential coefficient, 

-=— ■ = n(n — l)acc n - 2 ; 

and for the third, 

— - = n(n — 1) (n — 2)aa5*- 3 ; 

and for the fourth, 

d*u 

— = /i(/i - 1) (^ 2) (w — 3)a^- 4 . 



SEC. IY.J SUCCESSIVE DIFFERENTIALS. 125 

It is plain, that when n is a positive integral number, 
the function 

u = ax n ) 

will have n differential coefficients. For, when n differ- 
entiations have been made, the exponent of x in the 
second member will be ; hence, the nth differential co- 
efficient will be a constant, and the succeeding ones will 
be 0. Thus, 

^ = n(n - 1) (n - 2) (n - 8) a. 1 

d n + l u 



Sign of the first differential coefficient 
72. If we have a curve whose equation is, 

y = A*)> 

and give to X any increment A, we have (Art. 13), 

y' -y __ /(s + K) - f{x) 
h - h ' 

and passing to the consecutive values, 

dy 

-— = tan a. 

ax 

If we so place the origin of co-ordinates that the curve 
shall lie within the ' first angle, h will be positive, and 
y' — y will be positive at all points where the curve 



126 



DIFFERENTIAL CALCULUS. 



[SEC. IV 



recedes from the axis of -Z", and negative where it ap 
proaches the axis; and this is true for consecutive as well 
as for other values. Hence, the curve will recede from the 
axis of X when the first differential coefficient is positive^ 
and approach the axis when that coefficient is negative. 

The general proposition for all the angles and every 
possible relation of y and #, is this : 

The curve will recede from the axis of JST when the 
ordinate and first differential coefficient have the same 
sign, and approach it when they have different signs. 



1. To determine whether a 
given curve, as AJBC, recedes 
from, or approaches to the axis 
of JT, at any point, as G : 
Find, from the equation of the 
curve, the first differential co- 
efficient, and see whether it is 
positive or negative. 




2. If the tangent becomes parallel to the axis of JT 
at any point, as J9, 



dy 
dx 



tan a z=z ; hence, a := 0. 



If the tangent becomes perpendicular to the axis of X y 
at any point, as A 



dy 
dx 



= tan a = oo ; hence, a = 90°. 



SBC. IV. J SUCCESSIVE DIFFERENTIALS. 



127 



Sign of the second differential coefficient. 

73. A curve is convex, towards the axis of abscissas 
when it lies between the chord and the axis ; and con- 
cave, when the chord lies between the curve and the axis. 

1. 





Figures (1) and (2) denote two curves, the one con- 
vex and the other concave towards the axis of Jf. 

Let PM be any ordinate of either curve, P'M' an 
ordinate consecutive with it, and P"M" an ordinate con- 
secutive with P'M'. 

If we designate the ordinate PM by y, P' Q' will be 
denoted by dy (Art. 2l), and we shall have, 

PM' = y + dy; 
and since P"M" is consecutive with P'M\ 
P"M" = y + dy + d{y + dy) 
= y + 2dy + d 2 y. 

MP + P"M" 



Since, 
hence, 
and 



MM' = M'M" =z dx, QM' = 



QM'- PM' = QF = 



d*y 



128 



DIFFERENTIAL CALCULUS. [SEC. IV. 



In the case of convexity, QM' > P'M f , and then^ 
d 2 y is positive. 





In the case of concavity, QM f < PM', and then, 
(Py is negative ; and since dx 2 is always positive, the 
second differential coefficient will have the same sign as 
the second differential of y. 

If we take the case in which the ordinates are nega- 
tive, the second differential coefficient will still have the 
same sign as the ordinate, when the curve is convex, 
and a different sign when it is concave. Hence, 

The second differential coefficient will have the same 
sign as the ordinate when the curve is convex towards 
the axis of abscissas, and a contrary sign when it is 
concave. 

1. The second differential of y is derived from dy in 
the same way that dy is derived from y (Art. 72) ; viz. : 
by producing the chord PP, and finding the difference of 
the consecutive values of P"Q" and SQ", which is P"S. 

The co-ordinates x and y determine a single point of 
the curve, as P ; these, in connection with dx and dy, 
determine a second point, P f , consecutive with the first ; 
and these two sets of values, in connection with the sec- 
ond differential of y, determine a third point, P', con- 
secutive with J* 



SEC. IV.] SUCCESSIVE DIFFERENTIALS. 129 

Hence, the co-ordinates x and y, and the first and second 
differential coefficients, always determine three consecutive 
points of a curve. 

2. When the curve is convex towards the axis of 
abscissas, the tangent of the angle which the tangent line 
makes with the axis of J5T, is an increasing function of 
x ; hence, its differential coefiicient, that is, the second 
differential of the function, ought to be, as we have found 
it, positive (Art. 19). 

When the curve is concave, the first differential coefii- 
cient is a decreasing function of the abscissas; hence, the 
second differential coefficient should be negative (Art. 19). 

Applications. 

■74. The equation of the circle, referred to its centre 

and rectangular axes, is, 

x* + y 2 = H 2 ; hence, ^ = - ' -. 

ax y 

x 

Placing = 0, we have, x — 0. 

if 

Substituting this value of x in the equation of the circle, 
we have, 

y = ±^; 

hence, the tangent is parallel to the axis of abscissas at 
the two points where the axis of ordinates intersects the 
circumference. 

dy x 

y 

substituting this value in the equation of the circle, 
x — -±= H ; hence, 



If we make, ~ = — - ; = <x> , we have, y = ; 



130 DIFFERENTIAL CALCULUS. [SEC. TV. 

the tangent is perpendicular to the axis of abscissas at 
the points where the axis intersects the circumference. 

1. For the second differential coefficient, we find, 

d 2 y IP 

dx ? ~ y 3 ' 

which will be negative when y is positive, and positive 
when y is negative. Hence, the circumference of the 
circle is concave towards the axis of abscissas. 

2. If we apply the same process to the equation of 
the ellipse, of the parabola, and of the hyperbola, we 
shall find that the tangents, at the principal vertices, are 
parallel to the axes of ordinates ; that the second differ- 
ential coefficient and ordinate, in all the cases, except that 
of the opposite hyperbolas, have contrary signs ; and hence, 
all these curves, except the conjugate hyperbolas, are con- 
cam toivards the axis of abscissas. 

maclaurin's THEOREM. 

■75. Maclaurin's Theorem explains the method of de* 
veloping into a series any function of a single variable. 
Let u denote any function of x, as, for example, 

u = (a + x) m (1.) 

It is required to develop this, or any other function of 
#, into a series of the form, 

u = A + Bx + Cx 2 + Dx 3 + Ex± + &c. . . (2.) 

in which A, B, (7, D, &c, are independent of #, and 
arbitrary functions of the constants which enter into the 



SEC. IV.] MACLAURIN'S THEOREM. 131 

second member of Equation ( 1 ). When these coeffi- 
cients are found, the form of the series will be known. 

Since the coefficients, A, B, C, &c, are, by hypothesis, 
independent of cc, each will have the same- value for 
x = 0, as for any other value of x\ hence, it is only 
necessary to determine them for x = 0. 

If we make x = 0, in Equation (2), all the terms 
in the second member, after the first, w T ill become zero, 
and the second member will reduce to A, which is what 
the function u becomes in Equation ( 1 ), when x = 
That value is thus indicated : 

(w)„_o = -*• 

If we find the successive differential coefficients of u % 
from Equation (2), we shall have, 

^ = B + 2 Ox + SBx 2 + ±Ex 3 + &c. 
ax 

d ^ ■ = 2(7 + 2.ZDx + ZAJEh? + &c. 
dx 2 

V-' = 2.3D + 2.3AMc + &c. 
ax 3 

&c, &c. ; 

whence A = (u) x ^ 

Xdx/x^o 

a -- U d --\ 

1.2\dx 2 / x -Q 



D = 






1.2.3 V dx 3 / x = o 



132 DIFFERENTIAL CALCULUS. | SEC. IV 

hence, 

" = w— + (*)._," + Hie)..." 

which is Maclaurin's Formula. In applying the formula, 
we omit the expressions x = 0, although tfAe coefficients 
are always found under this hypothesis. 

EXAMPLES. 

1. Develop {a + sc) m , by Maclaurin's Formula* 
A = a m , 
Z? = ( — J = m{a + x) m ~ l = ma m ~\ 

~ 1/ d 2 u\ m(m — 1) , N m(m — • ^ 

-> 1 /d%\ m (ra — 1) (m — - 2) , , x 

1.2.3\e& 3 / 1 2 3 * WT *I 

__ m(m - l)(m - 2) 3 
- T 2 ~3~~~ a ' 

&a, &a, &c. 

Substituting these values in Equation (2), we have, 

m \T¥h 1) 

(a + x) m = a m + ma m - l x + T - - — 'a m ~ 2 x 2 

the same result as found by the Binomial Formula. 



SEC. IV.] maclaurin's theoeem, 133 

2. If the function is of the form, 

u r= -4— = (a + a)" 1 = a~4\ + -)" 1 . 
a 4- as v y \ at 

we find, 

a 

7 _ 1/^M - 1 X - 2 (a + g)- 3 J_ 

2.3U 3 Lo " 2.3 ~~ a 4 ' 

&a, &a, <fcc. 

Substituting these values in Maciaurin's Formula, 

1 1 xx 2 x 3 x 5 x 1 , o_ 

— ; — = - 9 H — i , H — -, H + <fcc. 

a -\- x a a 2 a 3 a 4 a 6 a 8 

3. Develop into a series, the function, 

u = V« 2 + » 2 = «(l + J) 

4. Develop into a series, the function, 
= y^-tff = «*(l - ^) 



v,2\ 2 



2 

,2\3" 



W 



Note. 76. Maciaurin's Formula has been demonstrated 
under the supposition, that in Equation ( 2 ) the coef- 
ficients are independent of cc, and that the equation is 



134 DIFFERENTIAL CALCULUS. [SEC. IV. 

true for every possible value that can be attributed tQ 
x. If, then, the function u becomes infinite, when x — 0, 
the equation cannot be satisfied ; neither can it be, if 
any one of the differential coefficients becomes infinite. 
Hence, any form of the function which produces either 
of these results, is excluded from the formula of Mac- 
laurin. The functions, 

u — log a;, u = cot sc, u = ax 2 , 

are examples of such functions. In the first case, 
u = — oo , when x — ; * in the second, u = oo , 
when x = ; and in the third, Z?, and the succeeding 
differential coefficients, become infinite, when x — 0. 



taylor's theorem. 

77. Taylor's Theorem explains the method of develop- 
ing into a series any function of the sum or difference 
of two independent variables. 

78. Since the sum or difference of two independent 
variables may always be denoted by a single letter, any 
function of the form, 

u' = f(x ± y% 

may be put under the form, 

u' = f(z), by making z — x =±= y. 

If we suppose z to be the abscissa, and u f the or* 
dinate of a curve, and give to z an increment A, z will 

* Bourdon, Art. 235. University, Art. 186. Legendre, Trig., Art. 22 



SEC. IV.] TAYLOK'S THEOREM. 135 

become z + A. If we pass to consecutive values, dz = dx, 
and 

du' du r , . x 

— = — = tan a. (Art. 13.) 

If we suppose x to remain constant, and y to receive 
the increment A, z will again become z + A, and when 
we pass to consecutive values, 

du' du f 

-=- — -=— = tan a. 
as ay 

Hence, in any function of the sum or difference of two 
independent variables, the partial differential coefficients 
are equal (Art. 32). 

79. As an example, take, 

u' = (« + y) n . 

If we suppose a; to vary, the first partial differential 
coefficient is, 

du r , x 

- = n(x + y)>-\ 

If we suppose y to vary, it is, 

du' , x 

— = n{x + y) n ~ l i 

and the same may be shown for the differential coef 
ficients of the higher orders. 

80. If any function of the form, 

««' = A x + y)> 

be developed into a series, it is plain that the series 



136 DIFFERENTIAL CALCULUS. [SEC. IV. 

must have terms containing the variables x and y, and 
that the constants, which enter into the given function^ 
must also enter into the development. Let us then 
assume, 

A u ') =A X + y)=A + J3y«+ Cy h + Dxf + &c. (I.) 
in which the terms are arranged according to the ascend- 
ing powers of y, and in which A, JB, (7, Z>, &c, are 
independent of y, but functions of a?, and arbitrary func- 
tions of all the constants which enter the primitive func- 
tion. It is now required to find such values for the ex- 
ponents a, 5, c, &c, and for the coefficients A, B, (7, D, &c, 
as shall render the development true for all possible values 
that may be attributed to x and y. 

In the first place, there can be no negative exponents. 
For, if any term were of the form, 



it might be written, 



-By—, 

B 



and making y = 0, this term would become infinite, 
and we should have, 

U ' = f[X) = 00 , 

which is absurd, since the function of cc, which is inde- 
pendent of y, does not necessarily become infinite when 

v = o. 

The first term A^ of the development, is the value 
which the primitive function u f assumes when we make 

y = o. 



BEC. IV.J TAYLOK'S THEOKEM, 137 

If we designate this value by u, we shall have, 

u = f{x). 

If we differentiate Equation ( 1 ), under the supposition 
that x varies, the partial differential coefficient is, 

du' dA dB dC . dD 

and if we differentiate, regarding y as a variable, the 
partial differential coefficient is, 

du' 

j- = aBy*- 1 + bCy h ~ l + cDy c ~ l + <fcc. . . (3.) 

But these differential coefficients are equal to each other 
(Art. 78) ; hence, the second members of Equations ( 2 ) 
and ( 3 ) are equal. Since the coefficients are inde- 
pendent of y, and the equality exists whatever be the 
value of y, it follows that the corresponding terms in 
each series will contain like powers of y, and that the 
coefficients of y in these terms will be equal.* Hence, 

a — 1=0, 6 — 1 = ' a, c — 1 = 5, &a, 

and consequently, 

a = 1, 6 = 2, c = 3, &c. 

Comparing the coefficients, we find, 

^ dA n IdB n ldC 

<fo ' 2 dfe 3 da; 

* Bourdon, Art. 195. University, Art. 178. 



138 DIFFERENTIAL CALCULUS. [SEC. IV. 

Since we have made, 

J\x + y) = u\ and J\x) = A = u, 

we shall have, 

A — u. b — — C — J) = 



dx' 1.2dx 2 ' 1.2.3ch?' 

and consequently, 

, , du d 2 u y 2 , d 3 u y 3 „ 

which is the formula of Taylor. 

In thfs formula, u is what u' becomes, when y ±= 0; 

du , d^' , . c? 2 w , <tf 2 ?/ 

— , what -=— becomes when y = ; -=-j > what -7-^- 

becomes when y = ; and similarly for the other coef- 
ficients. 

1. Let it be required to develop 

u' = J\x + y) = {x + y) n , 

by this formula. 

We find, 

du . c? 2 w , . o ■ o 

& = a n , -=- = n . <c n - K -j-z = n An — l)<c n - 2 + &c. : 
<fc dx 2 v 

hence, 

91\Yl 1 I 

w' = (a + y) n = x n -f nx n ~ l y -\ — -x n ~ 2 y* 

, w(w — 1) (w . — 2) o 

■ 1.2.3 y 



SECTION V. 



MAXIMA AND MINIMA. 

81. A maximum value of a variable function is greatei 
than the consecutive value which precedes, and the con- 
secutive value which follows it. 

A minimum value of a variable function is less than 
the consecutive value which precedes, and the consecutive 
value which follows it. 

If we denote any variable function by a, and the inde- 
pendent variable by cc, every relation between u and x 
will be denoted by the co-ordinates of a curve whose 
equation is (Art. 10), 

u = f(x). 

Let u r denote the consecutive or- 
dinate which precedes u, and u" the 
consecutive ordinate which follows it. 
Then, if u is a maximum, 



u > u\ 



and 



u > u" ; 



the curve therefore ascends just before the ordinate reaches 
a maximum value, and descends immediately aftericards • 
hence, at the point of maximum, it is concave towards 
the axis of abscissas (Art. 73). 

Since the curve ascends just before the ordinate reaches 
the maximum value, the first differential coefficient will 
be positive ; and since it then descends, the first differ- 
ential coefficient will be negative immediately after the 



140 



DIFFERENTIAL CALCULUS. [SEC. V. 



maximum value (Art. 72). Hence, at the point of 
maximum value of the ordinate, the first differential co- 
efficient will change its sign, and therefore passes through 0. 
Since the curve is concave towards the axis of absciims, 
the second differential coefficient is negative (Art. 73) ; 
hence, the conditions of a maximum value of u are, 



du 
dx 



= 0, 



dP'U 
and — , negative. 



§2. Denoting the consecutive or- 
dinates, as before, by u\ u, u", if 
u is a minimum, 



u < u\ 



and 



u < u" ; 



the curve, therefore, descends just before the ordinate 
reaches a minimum, and ascends immediately afterwards; 
hence, at the point of minimum, it is convex towards 
the axis of abscissas. 

Since the curve descends just before the ordinate reaches 
the minimum value, the first differential coefficient will be 
negative; and since it then ascends, the first differential 
coefficient will be positive immediately after the minimum 
value (Art. 72). Hence, at the point of minimum value 
of the ordinate, the first differential coefficient will change 
its sign, and therefore passes through 0. 

Since the curve is convex towards the axis of abscissas, 
the second differential coefficient is positive (Art. 73) ; 
hence, the conditions of a minimum value of u, are, 



du 



and 



cPu . . 

^, positive. 



SEC. V.] 



MAXIMA AND MINIMA. 



141 



83. Hence, to find the maximum or minimum value 
of a function of a single variable : 

1. Find the first differential coefficient of the function, 
place it equal to 0, and determine the roots of the equation. 

2. Find the second differential coefficient, and substi- 
tute each real root, in succession, for the variable in the 
second member of the equation ; each root which gives a 
negative result, will correspond to a maximum value of the 
function, and each which gives a positive result will cor- 
respond to a minimum value. 



/ 



Point of inflection. 

§4. A point of inflection is a point at which a curve 
changes its curvature with respect to the axis of ab- 
scissas. 

When a curve is concave towards 
the axis of abscissas, its second differ- 
ential coefficient is negative (Art. 72) ; 
when it is convex, the second differ- 
ential coefficient is positive (Art. 72) : 
therefore, at the point where the 
curve changes its curvature, the 
second differential coefficient changes 
its sign, and consequently passes 
through zero. 

In the first figure, the second differential coefficient, 
at the point M, changes from negative to positive ; in 
the second, from positive to negative. At the point M, 
in both figures, the first differential coefficient is equal 
to 0, and the tangent line separates the two branches 




142 DIFFERENTIAL CALCULUS. [SEC. V. 

of the curve. When the second differential coefficient is 
0, the ordinate at the point has neither a maximum nor 
a minimum. 

There are three consecutive points of the curve which 
coincide with the tangent, at the point of inflection. This 
is shown by the equality of the co-ordinates of the point 
M (in the curve and tangent), and of the first and sec- 
ond differentials. 

EXAMPLES. 

1. To find the value of x which will render the func- 
tion y a maximum or minimum in the equation of the 
circle, 

y +x - M : dx- y> 

X 

making, = 0,. gives, x = 0. 

*j 

The second differential coefficient is, 

When, x = 0, y = -Bj 



d 2 y 

dx* ~ 


x 2 + y 2 

y 3 


e, 


cPy 
dx* 



hence ' i§ = - i* 

which being negative, y is a maximum for H positive. 

2. Find the values of x which render the function y 
a maximum or minimum in the equation, 

y = a — bx + x 2 . Differentiating, 
-ir = - b + 2x, and -^ = 2 ■ 



SEC. V.J MAXIMA AND MINIMA. 143 

and making, — b + 2x = 0, 

b 

gives, x = -• 

Since the second differential coefficient is positive, this 
value of x will render y a minimum. The minimum 
value of y is found by substituting the value of x, in 
the primitive equation, It is, 

b 2 
V = «-■£• 

3. Find the value of x which will render the function 
u a maximum or minimum in the equation, 

u = a 4 + b 3 x — c 2 x 2 . 
cLu b 3 

a, = b3 - 2 ° 2x > hence ' x = §35 » 

(H 2 u 
and ^ = - 2c " 

hence, the function is a maximum, and the maximum 
value is, 

« = «* + —. 

4. Let us take the function, 

u = 3a 2 <c 3 — 6 4 <e + c 5 . 

•7?/ fi 2 

We find, 5P = 9a 2 a 2 - JV and a; = ± — . 

dx 3a 

The second differential coefficient is, 

a? = 1SaZx - 



144 DIFFERENTIAL CALCULUS. [SEC. V 

Substituting the plus root of #, we have, 

a* = + &ahz > 

which gives a minimum, and substituting the negative 
root, we have, 

which gives a maximum. 
The minimum value of the function is, 

u = c* — — ; 

9a' 



and the maximum value, 



u = c 5 4- — « 
9a 



5. Find the values of &, which make w a maximum 
or minimum in the equation, 

u = a; 5 — 5a; 4 + 5a? 3 — 1. 



. ( a; = 1, a maximum, 
f x = 3, a minimum. 



6. Find the values of a;, which make u a maximum 
or minimiun in the equation, 

v =* x 3 — 9x 2 + 15a; — 3. 

. j a; = + 1, a maximum. 

( aj = + 5, a minimum. 



SEC. V.] MAXIMA AND MINIMA. 145 

1. Find the values of ar, which make u a maximum 
or minimum in the equation, 

u = x 3 — Sx 2 + Sx + 1. 
Ans. There is no such value of aj, since the second differ- 
ential coefficient reduces to 0, for x = 1 ; hence, only 
one condition of a maximum or minimum is fulfilled,* 

§5. Notes. 1. In applying the preceding rules to 
practical examples, we first find an expression for the 
function which is to be made a maximum or minimum. 

2. If in such expression, a constant quantity is found 
as a factor, it may be omitted in the operation; for the 
product will be a maximum or a minimum when the 
variable factor is a maximum or minimum. 

3. Any value of the independent variable which renders 
a function a maximum or a minimum, will render any 
power or root of that function, a maximum or minimum; 
hence, we may square both members of an equation to 
free it of radicals, before differentiating. 

8. To find the maximum rectangle which can be in- 
scribed in a given triangle. 

Let b denote the base of the triangle, h the altitude, 
y the base of the rectangle, and x its altitude. Then, 

u — xy = the area of the rectangle. 
But, b : h : : y : h — x; 

bh — bx 



hence, y = 



h 



* We have limited the discussion to a single class of maxima and 
minima, viz. : that in which the first differential coefficient of the func- 
tion is 0, and the second negative or positive. 



14G DIFFERENTIAL CALCULUS. 

and consequently, 

bhx — bx 2 b /7 „ x 

u = T = T (hx - x 2 ); 



[sec. v. 



and omitting the constant factor — , we may write, 

u' = hx — x 2 ; 

for, the value of x 7 which makes -&' a maximum, will make 
u a maximum (Art. 85) ; hence, 



du' 
dx 



= h — 2x 7 



or, 



•^i 8 



therefore, the altitude of the rectangle is equal to half 
the altitude of the triangle ; and since, 

the area is a maximum (Art. 8l). 

9. What is the altitude of a cylinder inscribed in a 
given cone, when the volume of the cylinder is a maxi- 
mum? 

Suppose the cylinder to be in- 
scribed, as in the figure, and let 

AJB = a 7 BG=b, AZ> = x, ED = y\ 

then, JBD = a — x = altitude of 
the cylinder, and 

<xy 2 {a — x)* = volume = v . . (1.) 

From the similar triangles AED 
and ACB, we have, 

* Legendre, Bk. VIII. Prop, 2. 




SEC. V.] 



MAXIMA AND MINIMA. 



147 



x : y : : a : b ; whence, y 



bx 
a 



Substituting this value in Equation ( 1 ), we have, 

V = — 1 rX 2 (a — <c). 
a 2 y 

<7rS 2 
Omitting the constant factor — , we may write, 

v' — x 2 (a — x) ; 

for, the conditions which will make v' a maximum, will 
also make v a maximum (Art. 85). 

By differentiating, we have, 

-^- = 2ax — 3x 2 . 
ax 



Placing, 
we have, 

But, 



2ax — 3a 2 = 0, 
x = 0, and 



2 
x = -a. 



dx 2 



= 2a — 6x = — 2a. 



Hence, the cylinder is a maximum, when its altitude is 
one-third the altitude of the cone. 

10. What is the altitude of a 
cone inscribed in a given sphere, 
when the volume is a maximum? 

Denote the radius of the given 
sphere by r, and the centre by C. 
Let A be the vertex of the re- 
quired cone, BD the radius of its 

base, which denote by y, and denote the altitude AB 
by x. Then, 




148 DIFFERENTIAL CALCULUS. 

y 2 = 2rx — # 2 ;* 
and if we denote the volume of the cone by v, 

o = %nrx(2rx — x 2 ) = ±<x(2rx 2 — x 3 ).] 
Omitting the constant factor £#, we have, 



[sec. \ 



do' 

dx 



= 4rx — 3x 2 i hence, 



4rx — dx 2 = 0, 



and 



x = -r; 



that is, the altitude of the cone is four-thirds of the 
radius. 

11. What is the altitude of a cone inscribed in a sphere 



when the convex surface is a maximum ? 



Ans. - r. 



12. What is the length of the axis of a maximum 
parabola which can be cut from a given right cone with 
a circular base? 

Let BAG be a section of the 
cone by a plane passed through the 
axis ; and FD G a parabola made 
by a plane parallel to the element 
BA. 

Denote BG by S, AB by a, and 
CE by x ; then, BE = b — x, 
and FE ) the common ordinate of 
the circle and parabola, is equal to 
yjbx — x 2 .\ 

* An. G., Bk. II. Art. 4—8. f Le g«> Bfc Vm - Pr0 P- 5 - 

$ Lfegfcndre, Bk. IV. Prop. 28. Cor. 2. 




StC.Y.] MAXIMA AND MINIMA. HC 

By similar triangles, we have, 

b : a : : x : ? = D K 
o 

Hence, the area of the parabola (Art. 59) is, 

2 ax fa 

u = -z-j-yox — x 2 . 

3 

Omitting the constant factors, and remembering that the 
same value of #, which renders u a maximum, will render 
its square a maximum (Art. §5), and designating by u' 
the new function, we have, 

u r = x 2 (bx — x 2 ) = bx 3 — <c 4 , and 

V- = 3to 2 - 4«» ; or, a; = % ; and J9^ = ^2?, 
a# 4 4 

that is, the axis of the maximum parabola is three-fourths 
the slant height of the cone. 

13. What is the altitude of the maximum rectangle 
which can be inscribed in a given parabola? 

A?is. Two-thirds of the axis. 

14. "What are the sides of the maximum rectangle in- 
scribed in a given circle ? 

Ans. A square whose side is r-\/2. 

15. A cylindrical vessel, open at top, is to contain a 
given quantity f water. What is the relation between 
the radius of tne base and the altitude, when the inte- 
rior surface is a minimum? 

Ans. Altitude = radius of base, 



150 DIFFERENTIAL CALCULUS. [SEC. V 

16. Required the maximum right-angled triangle which 
can be constructed on a given line, as a hypothenuse ? 

Arts. When it is isosceles. 

17. Required the least triangle which can be formed 
by the tw r o radii, produced, and a tangent line to the 
quadrant of a given circle ? Arts. When it is isosceles. 

18. What is the altitude of the maximum cylinder 
which can be inscribed in a given paraboloid ? 

Ans. Half the axis. 

19. What is the altitude of a cylinder inscribed in a 
given sphere when its convex surface is a maximum? 

Ans. ri/2. 

20. What is the altitude of a cylinder inscribed in a 

given sphere, when its volume is a maximum ? 

2r 

Ans - V* 

21. Required the base of the maximum rectangle which 
can be inscribed in a given ellipse whose semi-axes are A 
and & Ans. Ay/2. 

22. A rectangular sheep-fold, to contain a given area, 
is to be built against a wall. Required the ratio of the 
least side to the larger, so that the cost shall be a min- 
imum. Aiis. 2, 

23. To circumscribe a given circle w r hose radius is r, 
by an isosceles triangle whose area shall be a minimum. 

Aiis. Perpendicular to base = 3r 



SECTION VI. 

DIFFERENTIALS OF TRANSCENDENTAL FUNCTIONS 
Differentials of Exponential and Logarithmic functions. 

§6, An Exponential function is one in which the in. 
dependent variable enters as an exponent; as, 

u = a* (1.) 

If, in a function of this form, we give to x an inere- 
ment A, we have, 

u' = a x + h = a x a h .... (2.) 

Subtracting Equation ( 1 ) from ( 2 ), member from mem* 
ber, we have, 

u' — u = a x a h — a* == a x (a h — 1) ; 
whence, - = a h — 1 (3.) 



Put, a = 1 4- 6, and develop by the binomial formula; 
we then have, 



152 DIFFERENTIAL CALCULUS. [SEC. VI. 

Substituting this value of a H , in Equation (3), and 
dividing by A, we have, 



u — u 
a x h 






If we now pass to consecutive values, by making A 
numerically equal to 0, we have, 

<fe . b 2 , Z> 3 &* J 5 . 



a*^ " 2 3 4 5 

and putting for 6 its value, a — 1, we have, 

Denoting the second member of Equation ( 4 ) by &, 
we have, 

— — = A;, or, du — da x = a'te . . (5.) 
a*dx v ' 

that is, the differential of a function of the form a x , is 
equal to the function, i?ito a constant quantity &, de- 
pendent on a, into the differential of the exponent. 

Relation between a and k, 

§7. The relation between a and k is very peculiar, 
and may be determined by Maclaurin's Formula, 

/ \ (du\ 1 (d 2 u\ , 1 (d 3 u\ , 
+ &c (6.) 



SEC. VI.] EXPONENTIAL FUNCTIONS. 153 

First, if we make x = 0, the function a x = 1 — lu\ 
The successive differential coefficients are found from 
Equation ( 5 ) ; viz. : 

d £ = a*k, and (*) =>; 
erf y- 1 = -=- = <?«*& = a x k 2 dx; hence, 
— -- a* and /— \ - &• 

= aip ' and (£) = ^ 



C? 3 W 

c&c 3 



&c, &c, &c. 

Substituting these values in Equation ( 6 ), we have, 

, kx k 2 x 2 , k 3 x 3 
u = a * = , + _ + _ + __ + 40. 

If we make # = -, we shall have, 

i- 111 

a* = 1-1 1 1 4- &c. ; 

T 1 1.2 1.2,3 T ' 

designating the second member of the equation by e, and 
employing twelve terms of the series, we find, 

e = 2.7182818. . . . ; 

i 
hence, a k = e, therefore, a = €* . . (7.) 

Equation ( 7 ) expresses the relation between a and k, 



154 DIFFERENTIAL CALCULUS. [SEC. VI 

A system of logarithms, called the Naperian system, 
has been constructed, whose base is, e = 2.7182818.... 
This, and the common system, whose base is 10, are the 
only systems in use. The logarithms, in the Naperian 
system, are denoted by £, and in the common system 
by log. We see from Equation ( 7 ), that k is the 
Naperian logarithm of the number a. If we take the 
common logarithms of both members of Equation (7), we 
shall have, 

log a — k log e (8.) 

The common logarithm of e = log 2.7182818 . . . . 
= .434284482 . . . . , is called the modulus of the common 
system, and is denoted by M. Hence, if we have the 
Naperian logarithm of a number, we can find the com- 
mon logarithm of the same number by multiplying by 
the modulus. 

If, in Equation (8), we make a = 10, we have, 

1 = k log e\ or, = = log e = M ; 

that is, the modulus of the common system is also equal 
to 1, divided by the Naperian logarithm of the common 

base. 

S8. From Equation (5), we have, 

du da* , , 
u a* 

If we make a = 10, the base of the common system, 
x = log w, and 

du 1 du -, 
<fc = — x = -- X Jf ; 

u k u 



SEC. VI.] LOGARITHMIC FUNCTIONS. 155 

that is, the differential of a common logarithm of a 
quantity is equal to the differential of the quantity di- 
vided by the quantity into the modulus. 

89. If we make a = e, the base of the Naperian 
system, x becomes the Naperian logarithm, of u, and h 
becomes 1 : see Equation (7) ; hence, M = 1 ; and 

du 

dx = — ; 

that is, the differential of a Naperian logarithm of a 
quantity is equal to the differential of the quantity di- 
vided by the quantity ; and in this system, the modulus 
is 1. 

90. Having found that Jc is the Naperian logarithm 
of a, we have from Equation ( 5 ), 

du = a x ladx\ 

that is, the differential of a function of the form a x , 
is equal to the function, into the Naperian logarithm 
of the base a, into the differential of the exponent. 

EXAMPLES. 

1. Find the differential of u = a x . 

du = a x I a dx. 

2. Find the differential of u == Ix. 

du = — = x" \7x, 
x 

Note. This case would seem to admit of integration 
by the rule of Art. 35 ; but that *ule applies to alge- 



156 DIFFERENTIAL CALCULUS. [SEC. VI. 

braic functions only, and this form is derived from a 
transcendental function. 

3. Find the differential of m = y*. 

lu = xly; hence, 

— = x~ + lydx: hence, 

«* y 

by clearing of fractions, and reducing, 

du = xy x ~ l dy + y x ly dx; 

that is, equal to the sum of the partial differentials 
(Art. 32). 

4. Find by logarithms the differential of u = xy. 

lu = Ix + ly\* hence, 

du dx dy ^ . , . 

— = 1 — -; and by reducing, 

u x y 

du = ydx + xdy (Art. 27). 

x 
6. Find by logarithms the differential of u = - • 

lu = Ix — ly\\ hence, by differentiating, 

du dx dy ., ,, ., . 

— = ; and by reducing, 

u x y 

* = "**-**' (Art. 29). 

* Bourdon, Art. 230. University, Art. 185. 
f Bourdon, Art. 231. University, Art. 185. 



SEC. VI-] LOG A KIT II MIC FUNCTIONS 157 

6. Find the differential of u = ll-±-°\. 

\a — x) 

, 2adx 

du = 



a* — x £ 



?. Find the differential of u = l(——==)- 

\Va? + x 2 ' 



7 a 2 dx 

du 



x(a 2 + x 2 ) 



8. Find the differential of u = (a x + l) 2 . 

du 5= 2a x (a x + l)?ad£& 

9. Find the differential of u = • 

a* + 1 

six (cl\ x 

iO. Find the differential of u = — = (-) • 

x* \x/ 

Differential forms which have known integrals. 

91. If we have a differential in a fractional form, in 
which the numerator is the differential of the denomina- 
tor, we know that the integral is the Naperian logarithm 
of the denominator (Art. 89). It frequently happens, 
however, that we have to deal with fractional differen- 
tials which are not of this form, but which, by certain 
algebraic artifices, may be reduced to it. We shall give 
a few examples of such reductions. 



158 DIFFERENTIAL CALCULUS. I SEC. VL 

dx 



Form 1. 



/■ 



x' ± or 



Put x 2 db a 2 = v 2 ; then, ccda = vdv. 

Add wfa; to both members; then, 

xdx + vdx = #<## + ^c?y ; hence, 

(a? + #)c?<b = v(dx + dv) ; whence, 

dx + e?y dsc dx . 

; hence, 



* + ^ ^ ya 2 ± a 2 

/dx + dv r dx 

But in the first member, the numerator is the differen 
tial of the denominator ; hence, 

u yx 2 db a 2 



Form 2. f 



dx 



yaj 2 ± 2ax 



Put \^x 2 ± 2acc = » ; then, « 2 db 2tf# = v 2 . 

Adding a 2 to both members, and extracting the square 
root, 

n. n * i vdv 

x ± a = yv 2 + a 2 ; hence, ax 



and 



•V^ 2 +- a 2 ' 



l/a 2 =fc 2acc y^ 2 + a 2 



SEC. VI.] LOGARITHMIC FUNCTIONS. 150 

But from the first form, 
J x/v 2 + a 2 



Substituting for v its value, and for ytP + a 2 , its value, 
f ^ = l(x ± a + y/& =fc 2ax). 

J <\/x 2 dr 2dX 



^ 2adx 2adx 

Form 3. — -; or, 



a* — x l x* — a* 



~. 2adx 2adx dx , dx* 
Mnce, — = -, ■ y- t r = — ■ h 



a 2 — x 2 " " (a + x) {a — x) " a + x a — as 

rl dx dx \ r dx r dx 

J \a -\- x a — x) ~ J a + x J a — x 

f 2adx fa + x\ 

Also, /"-*- = ,fe^*). 

J x 2 — a 2 \x + «/ 

(See Example 6, page 126.) 

_ 2aefc 

Form 4. — • 

, seya 2 ± a; 2 



Put «Ja 2 + a? 2 = v ; whence, a 2 + a 2 = v 2 ; hence, 

sc 2 = v 2 — a 2 , and xdx = vdv, or, da = • 

x 

* University, Art. 180. (?cc Art. 158.) 



160 DIFFERENTIAL CALCULUS. [SEC. VI 

2a 
Multiply both members by — - — ; we have, 

xya 2 + x 2 

r 2adx r 2adv , (v — a\ 

/ — r = /"l i = M — i — J; hence, 



/2adx j/V^ 2 + ^ 2 " a \ 

x^/a 2 + x 2 \-y/a 2 + ^ 2 + &' 

In like manner we should find, 



/2adx i a — y^ 2 ~~ ^ 2 \ 

xv^a? — x 2 \a + i/o* — # 2 ' 

Form 5. A 



:\Za? — x 2 \a + -y/tf 



lAM 1 



Put - = v: then, x~ 2 dx = — dv: and 

#- 2 tfe — dv n j, 

; first lorm. 



y/cP~+ x- 2 i/a? + v 2 



SEC. VI.] LOGARITHMIC FUNCTIONS. 16] 



TABLE OF FORMS. 



1. fa x ladx = a x (Ex. l.) 

2. f'—=jfax- l =lx ..... (Ex.2.) 

3 - f(vy x - l dy + y x iy x <fo) = y x . (Ex. 3.) 

4. /— iL= = *(« + V^±~«" 2 )- (Form 1.) 
47 y^± a 2 

5. f——JL= = l(x ± a + y/x 2 =±= 2aA (2.) 
^ y^ 2 ± 2aa 

^r^ = l y=$\ (Form 3 ° 

/• 2adx 7 /x — a\ m 

s. / • g^_ = ? /vg_£g-« \, (Form4) 

./ ccyV + x t V-v/a^ + a; 2 + a' 

r ladx .la — -Ja 2 — x\ 

9 - / r * 2 = l \~Zrf=y (Form 4,) 

«/ ay a 2 — x 2 \a + ya 2 — x 2 ' 

/ ^a*~+ a;- 2 \ as / 




H C A 



1G2 DIFFERENTIAL CALCULUS. [SEC VI. 

CIRCULAR FUNCTIONS. 

92. Let be the centre of a circle, 
A the origin of arc, and BO, DH any 
two consecutive ordinates. Draw BE 
parallel to 0A\ draw the radius OB, 
and denote it by 1. Denote the arc 
AB by 2, and suppose z to be the independent vari- 
able. Then, BD will be the differential of the arc AB; 
ED, the differential of the sine, and EB the differential 
of the cosine, which will be negative, since it is a de- 
creasing function of the arc (Art. 19). 

93. Since the triangles OBG and DEB, have their 
sides respectively perpendicular to each other, they will 
be similar ; * hence, 

OB : 00 : : BD : DE; or, 
1 : cos z : : dz : d sin z ; whence, 

d sin z = cos zdz (1.) 

94. Again, 1 : sins : : dz : — c?cos z\ whence, 

d cos z = — sin z dz . . . . ( 2.) 

95. Since, 

cos z = 1 — ver-sin 3, d cos z = — d ver-sin z ; 
hence, d ver-sin z = sin zdz . . . . (3.) 

* Legendre, Bk. IV. Prop. 21. 



SEC. VI.] 



CIRCULAR FUNCTIONS. 



163 



96. Again, tan z 



sin z 
cos z 



; hence, 



, . cos z d sin 2 — sin s d cos s - . 
a tan 8 = 9 (Art. 29). 

COS 2 Z v ; 

Substituting for d sin z and d cos z, their values from 
Equations ( 1 ) and ( 2 ), we have, 



c? tan z = 



cos 2 s 



(4.) 



By similar processes, we can find the differentials of 
the co-versed-sine, cotangent, secant, and cosecant, in terms 
of the other functions and the differential of z. 



97. Denote the sine of the arc AB by y, its cosine 
by sc, its versed sine by v, and its tangent by t. If 
we regard each of these as the independent variable, and 
the arc z as the common function, and find the values 
of z from Equations (1), (2), (3), and (-4), we shah 
have, 



dz = 



When radius = 1, 
dy 



V^y 2 



dx 
dz — 



dz 



dz = 



^2v — v 2 

dt 
1 + t 2 " ' 



(5.) 
(6.) 

(8.) 



When radius = r, 



efe = — 

dz = - 



r~ — y l 
rdx 



dz 



dz = 



y^ — x 2 
rdv 



\/2rv — v 2 

r 2 dt 
r 2 + t 2% ' ' 



(9.) 
(10.) 
(11.) 
(12.) 



164 l) I F FE R E N T I A L CALCULUS. [SEC. VI 

The differential of the arc, in terms of either of the 
other functions is easily found. 

93. The following notation is employed to designate 
an arc by means of any one of its functions. 

sin - %, denotes the arc of which u is the sine, 
cos-" 1 ^, denotes the arc of which u is the cosine, 
tan- 1 ^, denotes the arc of which u is the tangent, 

&c, &c, &c. 



If we denote the sine of an -arc by - , instead of y, as 
in Equation (5), we shall have, 

u , du , , u 

y — - , ay — — , and z — sm -1 -• 
u a" * a ' a 

Substituting these values in Equation ( 5 ), we have, 

du 
dz = ——= (13.) 

-\fa 2 — u 2 

u 

Denoting the cosine of the arc by -, and making like 

ci 

'substitutions in Equation (6), we have, 

dz = _^*L, (u.) 

<y/a 2 — u 2 

u 

Denoting the ver-sine of the arc by - , and making like 



substitutions in Equation ( 7 ), we have, 

di 
y2au— 



dz = — - .... (15.) 



SEC. VI.] CIRCULAR FUNCTIONS. 165 

Denoting the tangent of an arc by - , we have from 

Equation (8), 

7 adu / -, „ \ 

dz = , •• (16.) 

a? + u 2 



EXAMPLES. 

1. Differentiate the function, 

z = cos~ l (u\fT— u 2 ) . 
(- 1 + 2u 2 )du 



dz = 



2. Differentiate the function, 

a = sin- 1 - \2uyl — ^ 2 J . dz = 



3. Differentiate the function, 

z = tan- 1 -, <fe = ^— — — -£ . 

y y 2 + & 

4. Differentiate the function, 

z = coscc 8in *. 
Make, cos as = w, and sin a; = y ; 
then, s = w*, and, (Art. 00), 

cfe = u^ludy + yuy~ l du; 
also, du = — sinxdx, and rfy = cossctfas; 



166 DIFFERENTIAL CALCULUS. [ SEC. VI 

hence, dz = u* lludy + - du) 



mxh 



sin 2 x\ 



= co8x sinx \lco8X cosx )dx. 

COS XI 



Differential forms which have known Integrals. 

99. The first four equations in Art. 92 furnish us four 
forms, by taking the integrals of both members. Equations 
(5), (6), (7), and (8), are of the same form as Equations 
( 9 ), ( 10 ), ( 11 ), and ( 12 ), except that the radius is 
1 in the first set, and r in the second; hence, the arc 
z, in each equation of the second set, is r times as great 
as in the corresponding equation of the first set.* 

Forms (13), (14), (15), and (16), are modified forms 
of ( 5 ), ( 6 ), ( 7 ), and ( 8 ). They differ from them only 
in the symbol by which the function of the arc is denoted. 

TABLE OF FORMS. 



1. / cos z dz =z sin z + G. 

2. / — sin z dz = cos z + (7. 

8. / sin z dz = ver-sin z + C. 



4. / — r- = tan z + C. 

J eos 2 2 



COS 2 3 



5. f-Jy= = sin-iy + tf. 

J yj - y 2 

* Leg., Trig. Art. 30. 



BBC VI.] 

6. 

7. 



9. 



10. 



11. 



12. 



13. 



14. 



15. 



16. 



CIRCULAR FUNCTIONS. 

dx 



1G? 






y/\ —x 2 

dv 



-\/2v — v 2 
dt 



= cos- 1 ^ + C. 



ver-sin- 1 ^ + C. 



IttJ> =t*n-n + a 



r rdy 

i 

J r 2 + t 2 



-y/r& — y 2 

— rdx 
yffi — x 2 

rdv 
<\/2rv + v 2 

r 2 dt 



= sin" 1 !/ + C. 

= cos- 1 ^ + C. 
= ver-sin -1 ^ + C. 
= tan- 1 * + C. 






/ 

/ 
/ 



du . u 
— - = sin- 1 h 0. 

y/a 2 - u 2 a 



— du 

<y/a 2 — u 2 



= cob- 1 - + a 

a 



du • \ u \ n 

= ver-sin -1 - + 0. 
a 



<\/2au — u 



/ 



adu 

a 2 + v 2 



= tan- 1 - + G. 
a 



108 DIFFERENTIAL CALCULUS. [SEC. VI. 

Applications. 

100. We may readily find the relation between the 
diameter and the circumference of a circle from either 
of the first four equations of Art. 97. 

1. To find this ratio from Equation (5), which is, 

: yT"-^^ ; dy ~ yT^2 z ~~ l ~ y) 

Developing by the Binomial Formula, we have, 

dz n , 1 2 , 1 - 3 4 . 1A8 i , *r i. 

^ = ! + 2 V + U V + 2A£ y + &C ' ; WhenCGj 

1 -J q "1 O K 

/* = ■ = y + i^ 3 +2X5^ + 2Z#7 2/7 + &C - 

If we make s = 30°, of which the sine, y is ~ , * 
we have, 

1 1 1.3 1.3.5 . 

30° = I A A \- &C. 

2 ^ 2.3.2 3 ^ 2.4.5.2 5 ^ 2.4.6.7.2 7 T 

By multiplying both members of the equation by 6. 
and taking twelve terms of the series, we have, 

180° = * = 3.1415924, 
which is true to the last place, which should be 6. 



* L gendre, Trig. Art. 64. 



SBC. VI.] CIRCULAR FUNCTIONS. 1G9 

2. Find the ratio from Equation (8), which is, 
, dt dz 1 

dz = r+J>> or ' lt = T+T> = (i + *)-- 

Developing by the Binomial Formula, we have, 

dz 

j = 1 - t 2 + t* - t* + t 8 - &c. ; whence, 

dz = dt — t 2 dt + Pdt — t G dt + t 8 dt — &c. 

I dz ~ z = tan- 1 -* = t — - + — — ~z + — — &c. 
J 3 5 7 9 

This series is not sufficiently converging. To find the 
value of the arc in a more converging series, we employ 
the following property of two arcs, viz. : 

Four times the arc whose tangent is - , exceeds the 

, 5 

arc of 45° by the arc whose tangent is — - • * 

Zoo 

* Let a denote the arc whose tangent is — Then, Leg., Trig. 

5 
Art, 36., 

2 tan a 6 



tan 2a — 



tan 4a = 



1 - tan 3 a 12 ■ 

2 tan 2a 120 



1 - tan 2 2a 119 * 



The last number being greater than 1, shows that the arc 4a ex« 
ceeds 45°. Making, 

4a = A, 45° = B % 



170 



DIFFERENTIAL CALCULUS. [SEC. VI. 



But, tan- g) = i- - jL. + -*. - ^ + &c, 

^'(ris)^ 23 



+ ^To-^ 



7 + &c. 



hence 



arc 45° = < 



239 3(239) 3 T 5(239) 5 7(239)' 
4 \J ~ dJP + 5J 5 ~ 7?f 7 + / 

_(J_ l_ +— 1 L_ +) 

\239 3(239) 3 T 5(239) 5 7(239) 7 / 



Multiplying both members by 4, we find, 

180° = * = 3.141592653. 

the difference, 4a — 45° = -4 — B = 6, will have for its tangent, 

tan A — tan B 1 

tan b = tan^ - *) = TTT^Al^B = 2S9> 

1 
hence, four times the arc whose tangent is — , exceeds the an vf W 



by an arc whose tangent is rrr. 



SECTION VII. 

TRANSCENDENTAL CURVES CURVATURE RADIUS OP 

CURVATURE INVOLUTES AND EVOLUTES. 

Classification of Curves. 

101. Curves may be divided into two general classes ; 
J st. Those whose equations are purely algebraic ; and 
2dly. Those whose equations involve transcendental quan- 
tities. 

Those of the first class, are called Algebraic curves, and 
those of the second, Transcendental curves. 

The properties of the Algebraic curves have been already 
examined; it therefore only remains to explain the proper- 
ties of the Transcendental curves. 

Logarithmic Curve. 

102. A logarithmic curve, is a curve in which one 
of the eo-ordinates, of any point, is the logarithm of the 
other. The co-ordinate axis to which the lines denoting 
the logarithms are parallel, is called the axis of logarithms, 
and the other, the axis of numbers. 

If we suppose Y to be the axis of logarithms, then X 

will be the axis of numbers, and the equation of the curve 

will be, 

y = log x. 



172 



DIFFERENTIAL CALCULUS, 



[SEC. VII 




General Properties. 

103. Let A be the origin of a system of rectangular 
co-ordinates, X the axis of numbers, and Y the axis of 
logarithms. 

If we designate the base 
of a system of logarithms 
by a, we shall have,* 

a v = x, 

in which y is the logar 
rithm of x. 

If we change the value 
of the base a, to a', we shall have, 

y 

a — x, 

in which y is the logarithm of jc, to the base a'. It is 
plain, that the same value of jc, in the two equations, will 
give different values of y, and hence : Each system of 
logarithms will give a different logarithmic curve. 

If we make y = 0, we shall have,f x = 1 ; and since 
this relation is independent of the base of the system of 
logarithms, it follows, that : Every logarithmic curve zoill 
intersect the axis of numbers at a distance from the origin 
equal to 1. 

This abscissa is denoted by the line AE. 

We may find points of the curve from the generaj 
equation, * 

a y = x, 



* Bourdon, Art. 221. University, Art. I §3. 
f Bourdon, Art. 285. University, Art. 1§~6, 



SEC. VH.] TKANSCENDENTAL CURVES. 173 

even without the aid of a table of logarithms. For, if 
we make, 

1 3 1 p 

y = o, y = 2' v=s 5' yz= V ' 

we shall find, for the corresponding values of #, 

x = 1, x =<\/a, x = a^/a^ x = \Ja, &c. 

If we make a = 10, the curve will correspond to the 
common system of logarithms ; and if we suppose 
a = 2.7182818..., to the Naperian system. Both curves 
pass through the point E. 

Base > 1. 

104. If we suppose the base of the system of loga- 
rithms to be greater than 1, the logarithms of all numbers 
less than 1 will be negative;* therefore, the values of y, 
corresponding to all abscissas between the limits of x — 0, 
and x = AE = 1, will be negative; hence, these ordi- 
nates are laid off below the axis of JT. When x = 0, 
y = — oo. Therefore, when the base is greater than 1, the 
corresponding curve is QPEK '. The curve cannot extend 
to the left of the axis of Y", since negative numbers have 
no real logarithms.! 

Base < 1. 

105. If the base of the system is less than 1, the log- 
arithms of all numbers greater than 1 are negative; and 
of all numbers less than 1, positive. Under this supposi- 
tion, the curve assumes the position Q' P' EK. The parts 



* Bourdon, Art. 235. University, Art. 1§6. 
f Fourdon, Art. 235. University, Art. 1§6 



174 DIFFERENTIAL CALCULUS. [SEC. VII. 

of the curves EPQ, EP Q\ are concave towards the axis 
of abscissas; the parts EK, EK\ are convex; and both 
curves, throughout their whole extent, are convex towards 
the axis of Y. 

Asymptote. 
106. Let us resume the equation of the curve, 

y = log x. 

If we denote the modulus of a system of logarithms by 
M", and differentiate, we have (Art. 88), 

dx^.. dy M 

But, -~ denotes the tangent of the angle which the 

tangent line makes with the axis of abscissas; hence, the 
tangent will be parallel to the axis of abscissas when 
x — oo, and perpendicular to it, when x = 0. 

But, when x — 0, y = — oo ; hence, the axis of ordinates 
is an asymptote to the curve. The tangent which is par- 
allel to the axis of JS^ is not an asymptote; for, when 
x = oo , we also have, y = oo (Art. 50). 

Sub-tangent. 

107. The most remarkable property of this curve, is the 
value of its sub-tangent T r M\ estimated on the axis of 
logarithms. We have found, for the sub-tangent, on the 
axis of X (Art, 45), 

TR = ^y; 
dy y ' 



SEC. VII.] TRANSCENDENTAL CURVES. 175 

and by simply changing the axis, we have, 

T'R' = ^x = M (Art. 106); hence, 
ax 

The sub-tangent, taken on the axis of logarithms, is equal 
to the modulus of the system from which the curve is 
constructed. In the Naperian system, M = 1; hence, the 
sub-tangent is equal to 1, equal to AM In the common 
system, it is denoted by the number, .434284482 . . . 




108. If a circle JSTPG be rolled along a straight line, 
AL, any point of the circumference, as P, will describe 
a curve, which is called a cycloid. The circle NPG is 
called the generating circle, and P, the generating point. 

Since each succeeding revolution of the generating circle 
will describe an equal curve, it will only be necessary to 
examine the properties of the curve APBL, described in 
one revolution. We shall, therefore, refer only to this 
part, when speaking of the cycloid. 

If we suppose the point P to be on the line AL, at A, 
it will be found at some point, as L, after all the points 
of the circumference shall have been brought in contact 
with the line AL. The line AL will be equal to the 
circumference of the generating circle, and is called the 



I7G 



DIFFERENTIAL CALCULUS. 



[SEC. VII. 



base of the cycloid. The line BM, drawn perpendicular 
to the base, at the middle point, is called the axis of the 
cycloid, and is equal to the diameter of the generating 
circle. 

Transcendental Equation of the Cycloid. 

109. Let CJV be the radius of the generating circle. 
Assume any point, as A, for the origin of co-ordinates. 
Let us suppose that when the generating point has de- 
scribed any arc of the cycloid, as AP, that the point in 
which the circle touches the base has reached the point N. 




Through JV, draw the diameter iV6r, of the generating 
circle : it will be perpendicular to the base AL. Through 
P. draw PR perpendicular to the base, and PQ parallel 
to it. Then, PR — NQ will be the versed sine, and PQ 
the sine of the arc JSFP to the radius CJV. Pat, 



CN = r, AR 
we shall then have, 



PR = NQ = y, 



PQ = ^/2ry - y\ x = AN - RN ' = arc NP - PQ, 
hence, the transcendental equation is, 



x = ver-sin l y — y2ry— y 2 



S&C*. VII,] TRANSCENDENTAL CURVES. 177 



Differential Equation. 

110. The properties of the cycloid are most easily 
deduced from its differential equation. This is found by 
differentiating both members of the transcendental equation. 
We have (Art. 97), 

^(ver-sin-fy) = ^ ; and 

V2ry - y 2 

d(— \/2ry — y 2 ) = — 7 ^ "" J^g . hence, 

y 2ry — y 2 

\%ry V 2 s/2vy y 2 y2ry — y 2 

which is the differential equation of the cycloid. 



Sub-Tangent, Tangent, Sub-Normal, Normal. 

111. If we substitute in the general equations of Arts. 

45, 46, 47, and 4§, the value of -^, found in the dif- 

dx 

ferential equation of the cycloid, we shall obtain the values 

of the sub-tangent, tangent, normal, and sub-normal. 

TR = —i= J- = sub-tangent; 

y 2ry — y 2 

TP = ^VJ^_ = tangent; 

v 2r y - y 2 

PN = -yjlry = normal; 



RN = y/2ry — y 2 — sub-normal. 



178 



DIFFERENTIAL CALCULUS. [SEC. VH. 




These values are easily constructed, from their connec- 
tion with the parts of the generating circle. 

The sub-normal PJV, for example, is equal to PQ of 
the generating circle, since each is equal to ^/2ry — y 2 , 
hence, the normal PJSF, and the diameter GJV, intersect 
the base of the cycloid at the same point. Now, since 
the tangent to the cycloid at the point P must be per- 
pendicular to the normal, it will coincide with the chord 
PG of the generating circle. 

If, therefore, it be required to draw a normal, or a tan- 
gent, to the cycloid, at any point, as P, draw any line, 
as ng, perpendicular to the base AL, and make it equal 
to the diameter of the generating circle. On ng, describe 
a semi-circumference, and through P draw a parallel to 
the base of the cycloid. Through p, where the parallel 
cuts the semi-circumference, draw the supplementary chords 
pn, pg, and then draw through P the parallels PN, 
PG; and PiV will be a normal, and PG a tangent to 
the cycloid at the point P. 



Position of Tangent. 
112. The differential equation of the curve, 

ydy 



dx = 



yfery - y 2 ' 



SEC. VII.] TRANSCENDENTAL CURVES. 178 

may be put under the form, 



dy _ -/2ry - y 2 _ fet _ ^ 



dx ' y My 

If we make y = 0, we shall have, 

da; ' 

and if we make y — 2r, we shall have, 

ft''- 0- 

hence, the tangent lines drawn to the cycloid at the points 
where the curve meets the base, are perpendicular to the 
base ; and the tangent drawn through the extremity of the 
greatest ordinate, is parallel to the base. 

Curve Concave. 
113. If we differentiate the equation, 

dx = v d y 

y/2ry — y 2 
regarding dx as constant, we obtain, 

z ydyfrdy - ydy) 

o = j/a^r - jr (** + cif) - J ^=z=r-> 

or, by reducing and dividing by y, 

= (2ry — y 2 )d 2 y + rdy\ 
whence we obtain, 

#y = - *»* ■ * 

9 2ry — y 2 ' 

and hence, the curve is concave towards the axis of 

abscissas (Art. "73). 



180 



DIFFERENTIAL CALCULUS. [SEC. VTL 



Area of the Cycloid. 

Ill, The area of the cycloid may be found in a very 
simple manner, by constructing the rectangle AFBM, and 
considering the portion AFB. 

If we regard F as KB 

an origin of co-ordinates, 
FB as a line of ab- 
scissas, and take any 
ordinate, as 




we shall have, 
But, zdx 



KH — z — 2r — y, 
d(AHKF) = zdx. 

(<2r - y)ydy 



^2ry 



T 



= dy^/2ry~- y*\ 



whence, AHKF = fdy^/2ryT— y 2 + O. 

But this integral expresses the area of the segment of 
a circle, whose radius is r, and versed-sine y (Art. 99), 
that is, of the segment MIGF. If now, we estimate the 
area of the segment from i!/", where y = 0, and the area 
AFKH from AF, in which case the area AFKH = 0, 
for y = 0, we shall have, 

AFKH = MIGF; 

and taking the integral between the limits y = and 
y = 2r, we have, 

AFB = semi-circle MIGB, 



and consequently, 

area AIIBM 



AFBM - MIGB. 



SEC. VII.] TRANSCENDENTAL CURVES. 181 

But the base of the rectangle AFB3I is equal to the 
semi-circumference of the generating circle, and the alti- 
tude is equal to the diameter ; hence, its area is equal 
to four times the area of the semi-circle MIGB ; there- 
fore, 

area AHBM — SMIGB ; hence, 

The area AHBL is equal to three times the area of 
the generating circle. 

Surface described by the Cycloid. 

115. To find the surface described by the arc of a 

cycloid when revolved about its base. 

The differential equation of the cycloid is, 

dx - ydy 



^2ry — y 2 



Substituting this value of dx in the differential equation of 
the surface (Art. 62), it becomes, 



ds = 



3. 

< 2iry2ry 2 dy 



V^ry- y 2 
Applying Formula (E), (Art. 170), we have, 

s = 2«y/2r\- \y^V*ry-y* f- -r f I^- =\' 

L 6 6 J y2ry — y 2 J 

But, 



182 DIFFERENTIAL CALCULUS. [SEC. VII 

hence, 



s 



= 2* V / 2r[- \y 2 V2ry - y l - ^r(2r - y) S J 4- C. 



If we estimate the surface from the plane passing through 
the centre, Ave have C = 0, since at this point s = 0, 
and y = 2r. If we then integrate between the limits 
y — 2r, and y = 0, we have, 

s = - surface = — — *r 2 ; hence, 



s = surface = — — irr 2 



that is, the surface described by the cycloid, when it is 
revolved around the base, is equal to 64 thirds of the 
generating circle. 

The minus sign should appear before the integral, since 
the surface is a decreasing function of the variable y 
(Art. 19). 

Volume generated by the area of the Cycloid. 

116. If a cycloid be revolved about its base, it is re- 
quired to find the measure of the volume which the are3 
will generate. 

The differential equation of the cycloid is, 

a = ydy 



y2r — y 2 
If we denote the volume by F", we have (Art. 69), 

i/2ry — y 2 



J. VII.] 



TEANSCENBENTAL CURVES. 



183 



If we apply Formula (E) (Art. 170), we shall find, after 
three reductions, that the integral will depend on that of 

dy 



V^T 



r 



But the integral of this expression is the arc whose versed 

y 
sine is - (Art. 99). Making the substitutions and reduc- 
tions, we find the volume equal to five-eighths of the 
circumscribing cylinder. 

Spirals. 

117. A Spiral, or Polar Line, is a curve described by 
a point which moves along a right line, according to any 
law whatever, the line having at the same time a uniform 
angular motion. 

Let ABC be a straight 
line wmich is to be turned 
uniformly around the point 
A. When the motion of 
the line begins, let us sup- 
pose a point to move from 
A along the line, in the 
direction AB C. When 
the line takes the posi- 
tion ABE, the point will 

have moved along it, to some point, as B, and will have 
described the arc AaB, of the spiral. When the line 
takes the position AB f JE\ the point will have described 
the curve AaBB\ and when the line shall have complet- 
ed an entire revolution, the point will have described the 
curve AaBB'B. 

If the revolutions of the radius-vector be continued, the 




184 DIFFERENTIAL CALCULUS. [SEC. VII, 

geu crating point will describe an indefinite spiral. The 
point A, about which the right line revolves, is called 
the pole ; the distances AD, AD\ AB, are called ra- 
dius-vectors or radii-vectores ; and the parts AaDD'B, 
BFF' C, described in each revolution, are called spires 

If, with the pole as a centre, and AB, the distance 
passed over by the generating point in the direction of 
the radius-vector, during the first revolution, as a radius, 
we describe the circumference BUE\ the angular motion 
of the radius-vector about the pole A, may be measured 
by the arcs of this circle, estimated from B. 

If we designate the radius-vector by u, and the meas- 
uring arc, estimated from B, by t, the relation between 
u and t, may be expressed by the equation, 

u = f{t), or u — at n , 

in which n depends on the law according to which the 
generating point moves along the radius-vector, and a on 
the relation which exists between a given value of u 9 and 
the corresponding value of t. 

General Properties. 

118. When n is positive, the spirals represented by 
the equation, 

u — at n , 

will pass through the pole A. For, if we make t = 0, 
we shall have, a = 0. 
But if n is negative, the equation will become, 

a 
u = at~ n ; or, u = — 



SEC. VII. J TRANSCENDENTAL CURVES. 185 

from which we shall have, 

for, t = 0, u = oo, 

and for, £ = oo , u = ; 

hence, in this class of spirals, the first position of the 
generating point is at an infinite distance from the pole : 
the point will then approach the pole as the radius-vector 
revolves, and will only reach it after an infinite number 
of revolutions. 

Spiral of Archimedes. 

119. If we make n = 1, the equation of the spiral 

becomes, 

u = at. 

If we designate two different radii-vectores by u' and 
u", and the corresponding arcs by t' and t'\ we shall have, 

u f = at', and u" = at'\ 

and consequently, 

u' : u" : : t' : t" ; that is, 

The radii-vectores are proportional to the measuring 
arcs, estimated from the initial point. 

This spiral is called the spiral of Archimedes. 

If we denote by 1, the distance which the generating 
point moves along the radius-vector, during one revolu- 
tion, the equation, 

u = at, 

will become, 

1 = at: or, 1 x - = $. 

a 



186 DIFFERENTIAL CALCULUS. [SEC. VII 

But since t is the circumference of a circle whose 

radius is 1, we shall have, 

- = 2-7T, and consequently, a = — -• 
a 2* 



Parabolic Spiral. 

120. If we make n = -, and a = i/tyi we have, 
for the general equation, 

u = '\plp x t 2 ; or, u 2 = 2pt y 

which is the equation of the parabolic spiral. 

If t = 0, u = ; hence, this spiral passes through 
the pole. 

Hyperbolic Spiral. 

121. If we make n = — 1, the general equation of 
spirals becomes, 

u = at- 1 ; or, ut = a. 

This spiral is called the hyperbolic spiral, because of 
the analogy which its equation bears to that of the hy- 
perbola, when referred to its asymptotes. 

If, in this equation, we make, successively, 

* = *' t = 2' t = 3' *- te 'J» &c, > 
we shall have the corresponding values, 

w = a, w = 2a, i* == 3a, w = 4a, &a 



SEC, VH.] TRANSCENDENTAL CURVES. 



18? 



Logarithmic Spiral. 

122. Since the relation between u and t is entirely 
arbitrary, we may, if we please, make, 

t = log u. 

The spiral described by the extremity of the radius- 
vector, under this supposition, is called the logarithmic 
spiral. 

Direction of the measuring arc. 

123. The arc, which measures the angular motion of 
the radius-vector, has been estimated from right to left, 
and the value of t regarded as positive. If we revolve 
the radius-vector in a contrary direction, the measuring 
arc will be estimated from left to right, the sign of t 
will be changed to negative, and a similar spiral will be 
described. 

Sub-tangent in Polar Curves. 

124. The Sub-tangent, in spirals, or in any curve, 

referred to polar co-ordinates, 
is the projection of the tangent 
on a line drawn through the 
pole, and perpendicular to the 
radius-vector passing through 
the point of contact. 

Let A be the pole, AN = 1, 

the radius of the measuring 

arc, P any point of the curve, 

TP a tangent at P, and AT, 



\ 




188 DIFFERENTIAL CALCULUS. [ SEa Vn < 

perpendicular to AP, the sub-tangent. Let AP r be a 
radius-vector, consecutive with AP, and PQ, an arc de- 
scribed from the centre A. 

Then, NN r = dt, and #P = du, and, since P$ is 
parallel to NN\ we have, PQ = udt. But the arc 
P$ coincides with its chord (Art. 43), and since Q is 
a right angle, the triangles PQP' and TAP are similar ; 
hence, 

AT : ^.P : : PQ : QP' ; therefore, 

Sub-tangent -4jT : u : : weft : c?w. 

u'^df ft 
Whence, Sub-tangent A T ~ —— — -t nJrX . 

du n 

125. In the spiral of Archimedes, we have, 

1 



n = 1, and a = 



t 2 
hence, AT = — « 

2*71' 



2*' 



If we make t = 2tf, circumference of the measuring 
circle, we shall have, 

AT — 2<7r, circumference of the measuring circle. 

After in revolutions, we shall have 

t = 2mtf, 
and consequently, 

AT = 2m 2 * — m.2mir ; that is, 

The sub-tangent, after m revolutions, is equal to in times 



SEC. VII.] TKANSCENDENTAL CURVES. 189 

the circumference of the circle whose radius is the radius- 
vector. This property was discovered by Archimedes. 

126. In the hyperbolic spiral, n = — 1, and the value 
of the sub-tangent becomes 

AT = — a ; that is, 
The sub-tangent is constant in the hyperbolic spiral. 

Angle of the Tangent and Radius-Vector. 

12 7. We see that, 

AT _ uM 
AP "~ da' 

denotes the tangent of the angle which the tangent line 
makes with the radius-vector. 

In the logarithmic spiral, of which the equation is 
t — log u, 

we have, dt = — M ; 

u ' 

AT udt 

hence, -p^ == —=- = M : that is. 

AP du ' ' 

In the logarithmic spiral, the angle formed by th 
tangent and the radius-vector passing through the point 
of contact, is constant; and the tangent of the angle 
is equal to the modulus of the system of logarithms. 

If t is the Naperian logarithm of u, M is 1 (Art. 89), 
and the angle will be equal to 45°. 



190 DIFFERENTIAL CALCULUS. | SEC. VII, 

Value of the Tangent. 

128. The value of the tangent, in a curve referred to 
polar co-ordinates, is, 



PT = X /AJP*+ AT Z = ux/l+* 



:JAP 2 + AT 2 = uyjl 



Differential of the Arc. 



du 2 



129. To find the differential of the arc, which we de- 
note by s, we have. 



PP' = jQP f2 + QP 



2. 

5 



or, by substituting for PP f , QP', and PQ, their values, 
when P and P' are consecutive, we have, 



dz = ^du 2 + u 2 dt 2 



Differential of the Area. 

130. The differential of the area ADP, when referred 

to polar co-ordinates, is not an elementary rectangle, as 

when referred to rectangular axes, but is the elementary 

sector APP\ The area of this triangle is equal to 

AP r x PQ 

- — • If we denote the differential by ds, we have, 

AP'x QP _ (u + du)udt m 

CCS T — ■- • 

2 2 

or, omitting the infinitely small quantity of the second 
order, ududt (Art. 20), 

u 2 dt 
ds = — , 



SEC. VII.] TRANSCENDENTAL CURVES. 191 

which is the differ en tial of the area of any segment of a 
polar line. 

Areas of Spirals. 

131* If we denote by s, the area described by the ra- 
dius-vector, we have (Art. 130), 

u 2 dt 
ds = — ; 

and placing for u its value, at" (Art. 117), 

a 2 t 2n dt , a 2 t 2n + l 
ds = — - — , and s = — - + 6. 

If n is positive, (7=0, since the area is 0, when t = 0. 
After one revolution of the radius-vector, t — 2*r, and we 
have, 

a 2(2f)2» + l 

S ~ 4rc + 2 ' 
which is the area included within the first spire. 
132. In the spiral of Archimedes, (Art. 119), 

a = — , and n = 1 ; 
2<7r ' 

hence, for this spiral we have, 



24^' 



which becomes - , after one revolution of the radius-vec- 

o 

If 

tor ; the unit of the number - , being a square whose 

o 

side is 1. Hence, the area included by the first spire, is 
equal to one-third of the area of the circle whose radius 
is the radius-vector, after the first revolution. 



192 DIFFERENTIAL CALCULUS. [s EC. VII 

In the second revolution, the radius-vector describes a 
second time, the area described in the first revolution ; 
and in any succeeding revolution, it will pass over, or re 
describe, all the area before generated. Hence, to find 
the area, at the end of the mth revolution, we must in- 
tegrate between the limits, 

t = (m — 1)2*, and t = m.2<ir, 

which gives, 

m 3 — (m — l) 3 

s = it. 

3 

If it be required to find the area between any two spires, 
as between the mth and the (m + l)th, we have for the 
whole area to the (m + l)th spire, 

(m + l) 3 — m 3 

it i 

3 ' 

and subtracting the area to the mth spire, gives, 

t m _|_ i)3 _ 2m 3 + (m — l) 3 
s=z i 1 _ _^ L.<t — 2mir, 

for the area between the mth and (m + l)th spires. 

If we make m — 1, we shall have the area between 
the first and second spires equal to 2?r ; hence, the area 
between the mth and (m + \)th spires, is equal to 
m times the area between the first and second. 

J 33. In the hyperbolic spiral, n = — 1, and we have, 

a 2 t~ 2 a 2 
d s — — — fit an( j s — . 

2 ' 2t 

The area s will be infinite, when t = 0, but we can 



SEC. VII. ] CURVATURE. 193 

find the area included between any two radii-vectores b 
and c, by integrating between the limits t = b and t = cs, 
which will give, 

134. In the logarithmic spiral, t = Zw ; hence, 

dt -- — 

^d, — = — ; 

hence, s = J ~o~ = T + ^> 

and by considering the area 5 = 0, when w = 0, we 

have (7=0, and 

w 2 

5 = —• 



CUEVATUEE, 

135. The Curvature of a plane curve, at any point, 
is the departure from the tangent drawn to the curve at 
that point. This departure is measured by the distance 
which a point, moving on the curve, departs from the 
tangent in passing over a unit of length, denoted by the 
differential of the arc. In the same circle, or in equal 
circles, the departure from a tangent, at any point, is 
always the same ; hence, the curvature of a circle, at all 
points, is constant. 

Curvature of a circle is inversely as the radius. 

136. Let O and C be the centres of two unequal cir 
cles, having a common tangent at P. If we suppose the 



194 



DIFFERENTIAL CALCULUS. 



[SEC. VII. 



arcs to be the independent variables, we can denote the 
differential of one arc by Pb, and 
the differential of the other, by 
an equal arc Pa. Then, having 
drawn bP and aD, and the sines, 
bb\ aa\ and recollecting that each 
arc is equal to its corresponding 
chord, (Art. 43), we have, by de- 
noting the radii by r and r',* 

Pb 2 = 2r.Pb\ and Pa 2 = 2r'.Pa'\ 

since the arcs are equal, and Pb' == db, and Pa f = ca, 

2r.db — 2r'. ca ; hence, 

1 1 




db 



ac 



that is, 



The curvature of a circle varies inversely as its radius ; 
hence, the reciprocal of the radius of a circle may be 
assumed as the measure of its curvature. 

Orders of Contact. 

137. If two plane curves have one point in common, 
there is one set of co-ordinates (which may be denoted 
by cc", y"), that will satisfy the equations of both curves. 
If the curves have a second point in common, consecutive 
with the first, they will have a common tangent, at the 
common point, and the first differential coefficients will also 
be equal (Art. 43) ; this is called, a contact of the first order, 
If the curves have a third point in common, consecutive 
with the second, the second differential coefficients will be 



* Legendre, Bk. IV. P. 23. 



SEC. VII.] CURVATURE. 195 

equal (Art. 73) ; this is called, a contact of the second 
order. 

Generally, two curves have a contact of the nth order, 
when they have a common point, and the first n succes- 
sive differential coefficients of the common ordinate, equal 
to each other. 

Oscillatory Curves. 

138. Ajst Osctjxatrix, is a curve which has a highei 
order of contact with a given curve, at a given point, 
than any other curve of the same kind. The osculatory 
circle is by far the most important of all the osculatrices ; 
for it is this circle which measures the curvature of all 
plane curves. 

Osculatory Circle. 

139. The general equation of a circle, referred to rect- 
angular co-ordinates (Bk. II., Art. 5), is, 

(a- <*) 2 + (y- ft) 2 = B 2 . . . (1.) 

in which a and (3 are the co-ordinates of the centre, and 
x and y the co-ordinates of any point of the curve. 

If we regard a, /3, and i?, as constants, and differen- 
tiate the equation twice, and then find the differential 
coefficients of the first and second order, we have, 

(2.) 
and, d 2 y dx 2 .... (3.) 



dy x — a 




dx y — ft 

d 2 y dx 2 . , 
dx 2 V — ft 


• . 



196 DIFFERENTIAL CALCULUS. [SEC. VH 

In Equation ( 1 ) there are three arbitrary constants, a, 
:\ and It ; and values may be assigned to these, at pleas- 
ure, so as to cause the circle to fullfil three conditions, 
and three only. 

If we have any plane curve whose equation is of the 
form, 

y = A") ; 

and find, from this equation, the first and second differ- 
ential coefficients, for any point whose co-ordinates are 
x", y'\ we may then attribute such values to a, /3, and 
It, as shall make, 

x = as", y z= y" ; also, 

^ w ^ d2 y - d2 y" 

dx ~ dfc"' dx 2 ~ dx" 2 ' 

As no further general relations can be established be- 
tween the differential coefficients of the circle and curve, 
this circle will be osculatory to the curve at the point 
whose co-ordinates are x'\ y" (Art. 138). Since the 
co-ordinates of a point, and the differential coefficients of 
the first and second order, determine three consecutive 
points (Art. 13), it follows that, the osculatory circle passes 
through three consecutive points of the curve, at the point 
of osculation. 

Limit of the Orders of Contact. 

140. It is seen that the highest order of contact which 
a circle can have with any curve, is denoted by the num- 
ber of arbitrary constants which enters into its equation, 
less 1 ; and the same is true for any other osculatrix. 



SEC. VII.] 



CURVATURE. 



197 



Although it is impossible to assign a higher order of con- 
tact, to a circle, than the second, yet, at the vertices of 
the transverse and conjugate axes of the conic sections, the 
conditions which make the circle osculatory, also make the 
third differential coefficient zero, and hence give a contact 
of the third order. In general, when the order of contact 
is even, and the curve symmetrical with the normal at the 
point of osculation, the conditions imposed will give a con- 
tact of the next higher order. 



Radius of Curvature. 

141. If we find the value of R from Equations (1), 
(2), and (3), we have, 

(dx 2 + di/Y 



B = ± 



do:d 2 i/ 



(4.) 



which is the general value for the radius of the osculatory 
circle. 

If we denote the arc by z, we have (Art. 52), 



whence, 



dz 

R = ± 



d^ 
dxd 2 y 



(5.) 



Measure of Curvature. 

142. The curvature of a curve, at airy point, is measured 
by the curvature of the os- 
culatory circle at that point; 
hence, it is the reciprocal of 
the radius (Art. 136). 

If we assume two points, 
P and /*, either on the 




198 DIFFERENTIAL CALCULUS. [SEC. VII. 

same, or on different curves, and find the radii r and r' 
of the circles which are oscillatory at these points, then, 

i : 

curvature at P : curvature at P : : - : -7 • 

r r 

143. To find the radius of curvature, at any point of 
a plane curve, whose equation is of the form, 

Differentiate the equation twice, and substitute the values 
of the first and second differentials in Equation (4); the 
resulting equation will indicate the value of H for that 
point. 

If we use the + sign, when the curve is convex toward 
the axis of abscissas, and the — sign when it is concave, 
the essential sign of H will be positive, when JR is an 
increasing function of x. 

Radius of Curvature for Lines of the Second Order. 

144. The general equation for lines of the second or 
der (Bk. V, Art. 42), is, 

y 2 — mx + nx 2 , 

which gives, by differentiation, 

(m 4- 2nx)dx , , _ [4y 2 + (m + 2nx) 2 ~\(7x? 

dy = * — -4 — '—, dx 2 + dy 2 = l * y A \ — 

2y * 4y 2 

2nydx 2 — {m 4- 2nx)dxdy [4ny 2 — (m + 2nx) 2 ]dx 2 



SEC. VII.] CURVATURE. 199 

Substituting these values in the equation, 

3. 

(dx 2 + dy 2 ) 2 



R = : - 



dxd 2 y 



_> r4(mcc + nx 2 ) + (ra + 2;i£c) 2 ]" 
we obtain, i2 = — — ; 

which is the radius of curvature in lines of the second 
order, for any abscissa x. 

145. If we make x = 0, we have, 

that is, in lines of the second order, the radius of curva- 
ture at the vertex of the transverse axis is equal to half 
the parameter of that axis. 

146. If it is required to find the value of the radius of 
curvature at the vertex of the conjugate axis of an ellipse, 
we make (Bk. V, Art. 42), 

2B 2 B* 

m = —7-, n = — -r^, and x = A. 
A A 2 

which gives, after reducing, 

A 2 
R = -= ; hence, 

The radius of curvature at the vertex of the conjugate 
axis of an ellipse is equal to half the parameter of that 
axis. 

14 7. In the case of the parabola, in which n — 0, the 
general value of the radius of curvature becomes, 

3 

(m 2 + kmxy 
2m 2 



200 DIFFERENTIAL CALCULUS. [SEC. VIL 

If we make x = 0, we shall have the radius of curvature 
at the vertex, equal to — , or one-half the parameter. 

148. If we compare the value of the radius of curvature 
(Art. 144), with that of the normal line found in Art. 49, 

we shall have, 

_, (normal) 3 

R = — 9 -\ that is, 

1 9 

- m 2 
4 

The radius of curvature, at any point, is equal to the 
cube of the normal divided by half the parameter squared; 
and hence, the radii of curvature, at different points of 
the same curve, are to each other as the cubes of the cor- 
responding normals ; and the curvature is proportional to 
the reciprocals of those cubes. 

Evolute Curves. 

149. An Evolute curve is the locus of the centres of 
all the circles which are oscu- 

latory to a given curve. The 
given curve is called the Invo- 
lute. 

If at different points, P, P, 
P", &c, of an involute, or given 
curve, normals, PC, P'C, &c, 
be drawn, and distances laid off 
on them, on the concave side of 

the arc, each equal to the radius of curvature at the 
point, then the curve drawn through the extremities G, 
C, C", &c, of these radii of curvature, is the evolute 
curve. 




SEC. VII.] INVOLUTES AND EVO LUTES. 201 

A normal to the Involute is tangent to • the Evolute. 

150. Resuming the consideration of the first three 
equations of Art. 139, and changing slightly the forms of 
( 2 ) and ( 3 ), we have, 

(x-*y+ (y - (By = m . . . (1.) 

(x — a)dx -f (y — f3)dy = ... (2.) 
dx 2 + dy 2 + (y - f3)d 2 y = . . . (3.) 

Equations (2) and (3) were derived from Equation (1), 
under the supposition that cc, /3, and jR, were arbitrary 
constants, and of such values as to cause the circle to be 
osculatory to a given curve, at a given point. 

If now, we suppose the osculatory circle to move along 
the involute, continuing osculatory to it, the five quanti- 
ties, is?, a, /3, y, dy, will all be functions of the indepen* 
dent variable as, and a and f3 will bo the co-ordinates of 
the evolute curve. 

If we differentiate Equations ( 1 ) and ( 2 ) under this hy- 
pothesis, we have, 

(x — oi)dx + (# — P)dy — (x — a)doi — (y — /3)e?/3 = i?c?i?, 

dx 2 + eft/ 2 + (y — fi)d 2 y — eforfee — dSdy = 0. 

Combining the first with Equation (2), and the second 
with (3), we obtain, 

- (y — P) d/3 - (x - oi)doi = RdR, . (4.) 

— dvdx — d/3dy = . . . (5.) 



202 DIFFERENTIAL CALCULUS. [SEC. VII. 

From the last equation we have, 

d,S dx , N 

-T- = - -j- (6.) 

act ay v ' 

But Equation (2) may be placed under the form, 

y-P = - £,( x - *)> or > - y = - ^( a - «) • (»•) 

Substituting for — — , its value — , we have, 

'-* = !<"-•> < 8 -) 

Since Equations (1) and (8) are the same under dif- 
ferent forms, they represent one and the same line. 

Equation (7) is the equation of a normal to the invo- 
lute at a point whose co-ordinates are x and y, and passes 
through any point whose co-ordinates are a and f3 (Art. 44). 
Equation (8) is the equation of a tangent to the evolute 
at a point whose co-ordinates are a and /3, and passes 
through any point whose co-ordinates are x and y (Art. 43) ; 
therefore, 

TJie radius of curvature which is normal to the invo- 
lute is tangent to the evolute. 



Evolute and radius of curvature increase or decrease by the 

same quantity. 

151. Combining Equations (2) and (6), we have, 

x ~ a = irfi - 0) • • • • ( 9 -) 



BBC. VII.] INVOLUTES AND EVOLUTES. 203 

Substituting this value of x — a in Equation ( 1 ), we 
have, after reduction, 

(ch? + J/3 2 \ 



*-«-F$F) = * • . (»»•) 



Substituting the same value in Equation (4), reducing, 
and squaring both members, we obtain, 

Dividing (11) by (10), member by member, and taking 
the root, 

yW 2 + d/3* = dR .... (12.) 

But since a and (3 are the co-ordinates of the evolute, 
if we denote this curve by s, we shall have (Art. 52), 

dR = dz, dR — dz = 0, d(R — z) = ; 

whence, JS — - z = a constant (Art. 17) ; 

which, if we denote by c, gives, 

R = z + c . . . . . (13.) 

Since the difference between R and z is constant, it fol- 
lows that any change in one, will produce a corresponding 
and equal change in the other. 

If we draw any two radii of curvature, as PC, P'C, 
and denote them by R and R\ and the corresponding 
arcs of the evolute by z and s', we have, 

i? = z + c, and R r = z f + c ; 

whence, R r — R = z' — z\ that is, 

2%6 difference hetween any two radii of curvature is 



204 DIFFERENTIAL CALCULUS. [SEC. VT1. 

equal to the arc of the evolute intercepted behceen fjieir 
extremities. 

If we make z — 0, and denote the corresponding value 
of H by r, we have, 

r — + c — c ; hence, 

The constant c, is equal to the radius of curvature passing 
through the origin of arc of the evolute. 

If we suppose C to be the origin of arc of the evolute, 
then, CP — r — c\ and any radius of curvature, as C P\ 
will be equal in length to the line C CP. If then the 
evolute be developed, or unrolled, as it were, about the 
movable centre of the osculatory circle, the other ex- 
tremity of the radius of curvature will describe the involute 
curve. 

Evolute of the Cycloid. 

152. Let us resume the equation for the radium of curv- 
ature (Art. 141), 

B = _ (da? + d y *y 

dxcPg ' ' ' " *' 

If, in this equation, we substitute the value of dx and 
d 2 y, found in Art. 113, we have, 

B = ^{ryf = 2^2rg . . . . (2.) 

hence (Art. Ill), The radius of curvature is double the 
normal ; therefore, when the generating circle moves from 
A towards J/, any radius of curvature, as PP\ will be 
double the normal PN. 



SEC. VII.] INVOLUTES AND EVOLUTES. 



205 




If, in Equation ( 2 ), we make y — 0, we have, B = ; 
If we make y = MB = 2r, we have, R ~ 4/\ 

That is, the radius 
of curvature is zero at 
the point A, and twice 
the diameter of the 
generating circle at B. 

Since the radius of 
curvature and evolute 
are both zero at the 
point A, and since they 
increase equally (Art. 

151), it follows that the length of the evolute AP'A' is 
equal to A' MB, or twice the diameter of the generating 
circle. 

When the point of contact, JV, shall have reached Jf, 
the point P, will have described the involute APB, and 
the point P', the evolute AP'A'. If we describe a circle 
on A'M — \A'B, it will be equal to the generating circle 
of the cycloid, and the two circles will touch each other 
at M. Draw A'X parallel to AL. 

If now we suppose the circle whose centre is (7, to roll 
along the base from M to A, and the circle whose centre 
is C, to roll from A 1 to X, keeping the centres C and C\ 
in a line perpendicular to the base AL, the point P, of 
the upper circle, will re-describe the involute BPA, and 
the point P', will re-describe the evolute A 1 PA. But 
since the generating circles are equal, and since they are 
rolled over equal bases, the curves generated will be equal; 
hence, the involute and evolute are equal curves. 



906 DIFFERENTIAL CALCULUS. [SEC. VIL 

The part of the involute beginning at A, is identical with 
the part of the evolute beginning at A'. 

Since the involute and evolute are equal, the length of 
the involute APJB, is equal to twice the diameter of the 
generating circle; or the length of the entire cycloidal arc 
APBL, is equal to the entire evolute AP'A'L, or to 
four times the diameter of the generating circle. 

Equation of the Evolute. 

153. The equation of the evolute may be readily found 
by combining the equations, 

R dx 2 + dy 2 dy(dx 2 + dy 2 ) 

y ~ P = &y~* x ~* - dx~dH/~~' 

with the equation of the involute curve. 

1st. Find, from the equation of the involute, the values of 

I and **, 

and substitute them in the last two equations ; there will 
result two new equations, involving a, /3, #, and y. 

2d. Combine these equations with the equation of the 
involute, and eliminate x and y; the resulting equation 
will contain a, /3, and constants, and will be the equation 
of the evolute curve. 

Evolute of the common Parabola. 

154. Let us take, as an example, the common parabola, 
of which the equation is, 

y 2 = mx. 



SEC. VII.] INVOLUTES AND EVOLUTES. 207 

We shall then have, 

dy m m 2 dx 2 

dx = V V = ~~ ~Ty*~' 

and hence, 

n 4y' 3 /4:y 2 + m 2 \ 4y 3 + m 2 y 4y 3 
y ~ ** ~m 2 \ 4y 2 / : : m 2 = " ~m 2 + V<i 

and observing that the value of x — a is equal to that 
of y — (3 multiplied by — -— , we have, 



4y 2 + m 2 



hence we have, 



„ 4v 3 , 2y 2 m 

— = -£- , and a; — a = s _- ; 

m 2 ' m 2 ' 

substituting for y its value in the equation of the involute, 

y = »V, 
we obtain, 

3 

— p = — -; x — a = — 2aj — - ; 
and by eliminating &, we have, 



?2 



16 / 1 \ 3 



= —foe - 

27m\ 



■ml 



which is the equation of the evolute. 
If we make (3 = 0, we have, 



« = |»; 



208 DIFFERENTIAL CALCULUS. [SEC. VII. 

and hence, the evolnte meets the axis of abscissas at a 
distance from the origin equal to half the parameter. 
If the origin of co-ordinates be 
transferred from A to this point, 
we shall have, 

1 



2 m ' 



and consequently, 




^ 27m 



The equation of the curve shows that it is symmetrica] 
with respect to the axis of abscissas, and that it does not 
extend in the direction of the negative values of a'. The 
evolute CC corresponds to the part AJP of the involute, 
and CC" to the part AJP \ Both are convex towards the 
axis of J5T, 



INTEGRAL CALCULUS. 



Nature of Integration. 

155. In the Differential Calculus, we have developed 
a system of principles, and given a series of rules, by 
means of which we deduce, from any given function, two 
others; the first of which is called the Differential co- 
efficient, and the second, the Differential (Art. 25). In 
the Integral Calculus, we have to return from the differ- 
ential, to the function from which it was derived. 

This operation, as a fundamental problem, involves the 
summation of a series of an infinite number of terms, each 
of which is infinitely small (Art. 56). No general rule for 
the summation of such a series has yet been discovered ; 
and hence, we are obliged to resort, in each particular 
case, to the operation of reducing the given differential to 
some equivalent one, whose integral is known (Art. 34). 

Forms of differentials having known Algebraic Functions. 

156. We have found (Art. 35), that every differentia] 
monomial of the form, 

Ax m dx, 



210 INTEGRAL CALCULUS. 

in which m is any real number, except — 1, may be im 
mediately integrated ; and when m = — 1, the differentia! 
becomes that of a logarithmic function, and its integral 
is A Ix (Art. 89). 

157. We have seen that every differential binomial of 
the form, 

(a + bx n ) m x n ~ l dx, 

in which the exponent of the variable without the paren- 
thesis is less by 1 than the exponent of the variable with- 
in, can be immediately integrated (Art. 41). 

158. We have seen that every function of the form, 

Xdx, 

in which X can be developed into a series in terms of 
the ascending powers of as, has an approximate integral 
which may be readily found (Art. 42). 

Forms of differentials having known Logarithmic Functions. 

159. Any function of the form, 

A — ) 

x 

in which the numerator is the differential of the denom- 
inator, can be immediately integrated, since the integral is 
equal to Alx (Art. 89). In Art. 91, we have given five 
other forms of differentials, whose corresponding functions 
are logarithms. 



INTEGRATION OF FRACTIONS. 211 

Forms of differentials having known Circular Functions. 

160. In Art. 99, we have found sixteen differential 
expressions, each of which has a known integral corre- 
sponding to it, and which, being differentiated, will of course 
produce the given differential. 

In all the classes of functions, any differential expression 
may be considered as integrated, when it is reduced to one 
of the known forms ; and the operations of the Integral 
Calculus consist, mainly, in making such transformations 
of given differential expressions, as shall reduce them to 
equivalent ones, whose integrals are known. 



INTEGRATION OF RATIONAL FRACTIONS. 

161, Every rational fraction may be written under the 
form, 

Px n ~ l + Qx n ~ 2 . . . + Rx+S 
Fx* + Q'x n ~ l . . . + B'x + S' ' 

in which the exponent of the highest power of the variable 
in the numerator is less by 1 than in the denominator. 
For, if the greatest exponent in the numerator was equal 
to, or exceeded the greatest exponent in the denominator, 
a division might be made, giving one or more entire terms 
for a quotient, and a remainder, in which the exponent of 
the leading letter would be less by at least 1, than the 
exponent of the leading letter in the divisor. The entire 
terms could then be integrated, and there would remain 
a fraction under the above form. 



212 INTEGRAL CALCULUS. 

EXAMPLES. 

1. Let it be required to integrate the expression, 

2 adx 

• 

x z — a 2 

By decomposing the denominator into its factors, we 

have, 

adx 2 adx 



x 2 — a 2 " (x — a) (x + a) 
Let us make, 

2adx (A B 



— = ( + )dx, 

a) \x — a x + a/ 



(x — a) (x + a) 

in which A and B are constants, whose values may be 
found by the method of indeterminate co-efficients.* To 
find these constants, reduce the terms of the second mem- 
ber of the equation to a common denominator; we shall 
then have, 

adx (Ax + Aa + Bx — Ba)dx 

(x — a) (x + a) (x — a) (x + a) 

Comparing the two members of the equation, we find, 

2a = Ax + Aa + Bx — Ba\ 

or, by arranging with reference to x, 

(A + B)x + (A — B — 2)a = 0; hence, 

A + B = 0, and (A - B - 2)a = 0; 

whence, -4 = 1, J5 = — 1. 

* Bourdon, Art. 194. University, Art. ISO. 



INTEGRATION OF FRACTIONS. 213 

Substituting these values for A and J5, we obtain, 
2adx dx dx 



x 2 — a 2 " x — a x + a 
integrating, we find (Art. §9), 
adx 



/ctdx 
— — — - '= l(x — a) — l{x + a) + C\ consequently 

J x 2 — a 2 \x + a! 



2. Find the integral of, 

Sx — 5 



dx. 



x 2 — 6x + 8 ' 

Resolving the denominator into its two binomial factors, 
(x — 2), and (x — 4), we have, 

Sx — 5 A B 

^-6^+8 ~ ^~2 + ^T7 ; henCe ' 

3a; - 5 ^ta; - 4 A + Bx - 2B 

x 2 — 6<b + 8 "^ a; 2 - 6a; + 8 ' 

by comparing the coefficients of as, we have, 

- 5 = - ±A - 2.5, 3 = A + B, 

1 1 

which gives, B = -, ^1 = — -; 

substituting these values, we have, 

/ *L^JL_ <fc = _ ! /"_*_ + » /•_*. + c 

J x l ■■ Qx + 8 2«/ aj — 2 2*/ a - 4 ^ 

7 1 

= 2 l0g(aJ ~ 4 ) ~ 2 l0g( * ~ 2) 4 ' a 



214 INTEGRAL CALCULUS. 

Hence, for the integration of rational fractions : 

1st. Resolve the fraction into partial fractions, of- which 
the numerators shall be constants, and the denominators 
factors of the denominator of the given fraction. 

2d. Find the values of the numerators of the partial 
fractions, and multiply each by dx. 

3d. Integrate each partial fraction separately, and the 
sum of the integrals thus found will be the integral 
sought. 

INTEGRATION BY PARTS. 

162. The integration of differentials is often effected 
by resolving them into two parts, of which one has a 
known integral. 

We have seen (Art. 27), that, 

d(uv) == udv -f vdu, 
whence, by integrating, 

uv = J udv +J vdu, 
and, consequently, 

/ udv = uv — j vdu. 

Hence, if we have a differential of the form Xdx, which 
can be decomposed into two factors P and Qdx, of 
which one of them, Qdx, can be integrated, we shall 

have, by making / Qdx == v, and P = u, 

J Xdx —JPQdx — J udv = uv - J vdu . (].) 
in which it is only required to integrate the term I vdu. 



INTEGRATION BY PARTS. 215 



EXAMPLES. 



1. Integrate the expression, x 3 dx\/a 2 + x 2 . 
This may be divided into the two factors, 
, cc 2 , and xdx-y/a 2 + x 2 , 
of which the second is integrable (Art. 41). 



Put, u == cc 2 , and dv = xdx^fa? + x 2 ; 

then, 

( a 2 _|_ /^2\2 

xdxya? + a; 2 = — — - — -— 
3 

Substituting these values in Formula (1), 

udv = cc 2 ( — ) — J- — — x 2xdx; 

and finally, 

3. 

xfidx 
2. Integrate the expression, — -• 

(a 2 - x 2 f 
The two factors are, #, and xdx(a 2 — x 2 ) 2 . 

■A 1 



U 



= a; ; c?y = xdx(a 2 — a; 2 ) 2 ; t? 



</(a 2 .— a 2 ) 



/ wc?y = _ : + / : ; whence, 

J t/o? - a; 2 J yo 2 - ^ 2 

— = -7=5=1 + sm_1 « ( Art - »»)• 



(a 2 - a; 2 ) 



f yV- a»5 



216 INTEGRAL CALCULUS. 

INTEGRATION OF BINOMIAL DIFFERENTIALS. 
Form of Binomial. 

163. Every binomial differential may be placed undei 
the form, 

x m ~~ l dx{a + bx n )P, 

in which m and n are whole numbers, and n positive ; 
and in which p is entire or fractional, positive or nega- 
tive. 

1. For, if m and n are fractional, the binomial takes 
the form, 

X JL 

x 3 dx(a + bx 2 )P* 

If we make x = z 6 , that is, if we substitute for #, an- 
other variable, 2, with an exponent equal to the least 
common multiple of the denominators of the exponents 
of x, we shall have, 

1 JL 

x 3 dx(a + bx 2 )P = 6z n dz(a + bz 3 )P, 
in which the exponents of the variable are entire, 

2. If n is negative, we have, 

x m ~ l dx(a + bx~ n )P, 
and by making x — -, we obtain, 

- z~ m + l dz(a + bz n )Pj 
in which n is positive. 

3. If x enters into both terms of the binomial, giving 

the form, 

x m ~ dx(ax r + fec 71 )^, 



BINOMIAL DIFFERENTIALS. 21? 

in which the lowest power of x is written in the first 
term, we divide the binomial within the parenthesis by 
sc r , and multiply the factor without by x r P ; this gives, 

x m +P r - l dx(a + bx n ~ r )P, 

which is of the required form when the exponent 
m + p r ~ \ is a whole number, and may easily be 
reduced to it, when that exponent is fractional. 



When a Binomial can be integrated. 

164. — 1. If jo is entire and positive, it is plain that the 
binomial can be integrated. For, when the binomial is 
raised to the indicated power, there will be a finite num- 
ber of terms, each of which, after being multiplied by 
x m - l dx, may be integrated (Art. 35). 

2. If m == n, the binomial can be integrated (Art. 41) 

3. If p is entire, and negative, the binomial will take 

the form, 

x m - l dx 



(a + bx n )p 9 
which is a rational fraction. 

Formula 4& 

For diminishing the exponent of the variable without the p«* 

renthesis. 

165. Let us resume the differential binomial, 



218 INTEGRAL CALCULUS. 

.If we multiply by the two factors, x n and x~ n , the 
value will not be changed, and we obtain, 

x m - n x n ~ 1 dx(a + bx n )p. 

Now, the factor x*~ l dx(a + bx n )P is integrable, what- 
ever be the value of p (Art. 41). Denoting the first 
factor, x m ~ n by u, and the second by dv, we have, 

du =z (m — n)x m - n - 1 dx. and v = —, f-=— ; 

V ; ' (p + l)nb ' 

and, consequently, /* , ' , 

^ J J x m ~ l dx(a + fe n )^ = 

a; m - n (a + te n )^ +1 m — n p • ■ ■' % , , 

But, fx m - n - l dx(a + to 1 )^- ^= 

fx m - n ~ l dx(a + bx n )p(a + bx n ) = 

a fx m - n ~ l dx(a + bx n )p + bfx m ~ 1 dx(a + bx n )P; 

substituting this last value in the preceding equation, and 
collecting the terms containing, 

fx m ~ 1 dx(a + bx n )P, 
we have, (l + - ^ +^j fa- 1 M<* + hx *) p ~ 

x m -*(a + bx n )P + l — a{m — n) fx m " n - 1 dx(a + bx n )P 
(p + \)nb ; 



BINOMIAL DIFFERENTIALS. 219 

whence, 

(A) fx m ' l dx(a + bx n )P = 

x m ~ n (a + bx n )P +} — a(ra — n) I x m ~ n ~ l dx{a + bx n )P 
b{pn + m) 

This formula reduces the differential binomial, 
fx m ~ l dx(a + bx n )f, to fx m - n ~ l dx{a + bx n )P; 
and by a similar operation, we should find, 

fx m - n ~ 1 dx(a+bx n )P J to depend on, fx m - 2n ~ 1 dx(a + bx n Y\ 

consequently, each operation diminishes the exponent of 
the variable without the parenthesis by the exponent of the 
variable within. 

After the second integration, the factor m — n, of the 
second term, becomes m — 2n; and after the third, 
m — f 3n, &c. If m is a multiple of n, the factor m — n, 
m — 2n, m — 3n, &c, will finally become equal to 0, and 
then the differential into which it is multiplied will disap- 
pear, and the given differential can be integrated. Hence, 
a differential binomial can be integrated, when the ex- 
ponent of the variable without the parenthesis plus 1, is 
a multiple of the exponent within. 

APPLICATIONS. 

166, We have frequent occasion to integrate differentia] 
binomials of the form, 

x m dx , , -i. 

- = xr dx(a 2 — x 2 ) 2 . 



ya? — x 2 



:<J20 



INTEGRAL CALCULUS. 



The differential binomial x m ~ l dx(a + bx n )P will assume 
this form, if we substitute, 



for m* 



fct 


a, 


u 


6 2 . 


U 


n, 


CC 


ft 



m+ 1; 

a*; 

- i; 

2; 

2 * 



Making these substitutions in Formula d&, we have, 



/ 



x m dx 
y« 2 — x' 



x m ~ l r~-z , a 2 (m — 1) n x m ~ 2 dx 

V a 2 — x 2 -\ * - y / — = ; 

m m J J~J^-~ x 2 



so that the given binomial differential depends on, 

/x m ~ 2 dx 
-y/a 2 — x 2 

and in a similar manner this is found to depend upon, 

x m ~ K dx 



/ 



v / « 2 ~— # 2 



and so on, each operation diminishing the exponent of x by 
2, If m is an even number, the integral will depend, after 



m 



operations, on that of, 



/dx , x . . . 

— — sin- 1 - (Art, 
\/a 2 — x 2 a 



. 99). 



binomial differentials. 221 

Formula 28. 
For diminishing the exponent of the parenthesis. 

167. By changing the form of the given differential 
binomial, we have, 

fx m - 1 dx(a + bx n )P = 

fx m ~ l dx(a + bx n )P~ l {a + bx n ) = 
a fx m - l dx(a + bx n )P~ l + b fx m+n - 1 dx(a + bx n )P~ l . 

Applying Formula ^ to the second term, and observing 
that m is changed to m + n, and p to p — 1, we have, 

fx m+n - 1 dx(a + bx n )P- 1 = 

x m (a + bx n )p — am I x m - 1 dx(a + bx n )P~ l 
b(pn + m) 

Substituting this value in the last equation, we have, 

(JB) fx m - l dx(a + bx n )P = 

x m (a + to 1 )? + pna I x m ~ l dx(a + to 1 )*- 1 



£>7l + »i 



• l 

"9 



in tvhich the exponent of the parenthesis is diminished by 
1, for each operation. 



APPLICATIONS. 

3 



1. Integrate the expression dx(a 2 + x 2 )~ 

The differential binomial x m ~ l dx{a + bx n )p will assume 



222 INTEGRAL OALCULUS. 

this form, if we make m = 1, a = a 2 , 6=1, w = 2, 
and p = §. 

Substituting these values in the formula, we have, 

3 cc(a 2 + x 2 ) * + Sa 2 fdx{a 2 + » 2 P 
y <&(a 2 + a? 2 ) 2 = ^ 

Applying the formula a second time, we have, 



x(a 2 + x 2 ) 2 a 2 ~ dx 



Jdx(a 2 + x 2 ) = j + -J ^ 2 + x2 

But we have found (Art. 9l), 



dx 






2. Integrate the expression, dx\/r 2 — x 2 . 

The first member of the equation will assume this form, 
if we make, m = 1, a = r 2 , b == — 1, rc == 2, and 
^> = i. Substituting these values in the formula, we 
have, 

/ dx^/r^~— x 2 = -cc(r 2 — # 2 ) 2 + - r 2 /* — r— ; 

2 2 J y/^2 _ rgZ 

whence, by substitution (Art. 99), 

dx x/r 2 — x 2 = - a?(r 2 — a 2 ) 2 + - r 2 sin - * - -f G. 



BINOMIAL DIFFERENTIALS. 223 

Formula <B . 

For diminishing the exponent of the variable without the 
parenthesis, when it is negative. 

168. It is evident that Formula &> will only diminish 
m — 1, the exponent of the variable, when m is positive. 
We are now to determine a formula for diminishing this 
exponent when m is negative. 

From Formula A,, we deduce, 

Jx m ~ n - l dx(a + bx n )P = 

af»- n (a + bx n )P+ l — b(m + np) fx m - l dx(a + bx n )P 
a(m — n) ' 

changing m, to — m + n, we have, 

(<§) j x- m - l dx{a + bx n )p = 

x~ m (a + bx n )P + l + b(m — n — np) fx~ m + n - l dx(a + bx n )p 



— am 



in which formula, it should be remembered that the nega- 
tive sign has been attributed to the exponent m. 



APPLICATIONS. 

1. Integrate the expression x~ 2 dx(2 — x 2 ) 2 • 

The first member of Equation ( <£ ) will assume this form, 
if we make m — 1, a — 2, b — — 1, n = 2, and 
p = — - § . Substituting these values, we have, 

fx-*dx(2 - a?)" 1 = - — ^ " X2) 2 +/(2 - *)"*** 



224 INTEGRAL CALCULUS. 

The differential term in the second member will be in- 
tegrated by the next formula. 

Formula 2EK 

For diminishing the exponent of the parenthesis when it is 

negative. 

169. It is evident that Formula 2S will only diminish 
p % the exponent of the parenthesis, when p is positive. 
We are now to determine a formula for diminishing this 
exponent when p is negative. 

We find, from Formula S$, 

Jx m ~ l dx(a+ bx n )v- 1 =r 

— aF(a + bx n )P + (?n + np) fx m ~ l dx(a + bx n )P 
pna ? 

writing for p 9 — p + 1, we have, 

(SB 1 ) j x m ~ l dx(a + bx n )~P — 

x m (a + bx n )-v +l — (m + n — ■ ?*jp) fx m - 1 dx(a + foc n ) _ ^ + 1 

wa(_£> — 1) 

When p = 1, jp — 1 == 0; the second member be 
comes infinite, and the given expression becomes a ra< 
tional fraction. 



applications. 



1. Integrate the expression, / dx(2 



x 2 ) \ 



BINOMIAL DIFFERENTIALS. 225 

The first member of Equation 2> will assume this form, 
if we make m = I, a — 2, 6 — — 1, n — 2, and 
£> =r — | . Substituting these values, we have, 

y <fc(2 - tf 2 ) 2 = -A_ _ 1— ; 

since the coefficient of the second term, in the formula, 
becomes zero. 

Returning, then, to the example under the last formula. 
\ve have, 

/.-<*(« - rff* = - --'(»-*£_* + f!fiLrjed - a 

2. By means of Formula 3S», we are able to integrate 
the expression, 

/7« 

when jp is a whole number. 

The general formula will assume this form, if we make 
m = 1, cc = 3, a = a 2 , 6 = 1, ^ = 2. 

Each application of the formula will reduce the expo- 
nent — p, by 1, until the integral will finally depend on 
that of 

^_ = ltan-i- + (7 (Art. 99). 
a 2 + z 2 a a 

Formula VS. 

When the variable enters into both terms of the binomial, 

IfO, Let it be required to integrate the expression, 

x^dx 7 . n .-l 

- = x^dx(2ax — x 2 ) . 



\/2ax — x 2 



226 INTEGRAL CALCULUS. 

The second member may be placed under the form, 

/q-L _i. 

x 2 dx(2a — x) . 

We apply Formula &, by making, 

m = ? + 2 1 w = *> i 9 = - 2 ' a = 2 ' 6 = - * i 

we shall then have, 

J x 2 dx(2a — #) 



2 



2/ 



If we observe that, 

x * = x x , and # = # # 2 , 

and pass the fractional powers of x within the parentheses, 
we shall have, 

(|g) / x9 dx = 

J Y'2ax — x 1 



x<i- l <\/2ax — x 2 ( (2q — l)a f x<L~ l dx 



+ 



9 9. J ^/2ax - x 2 

Each application of this formula diminishes the expo- 
nent of the variable without the parenthesis by 1. If q 
is a positive and entire number, we shall have, after q 
reductions, 

f—7— _ = ver-sin- 1 - + C (Art. 99). 



K 



1 r 3 W~ 




1 



I 



I 

1 



